[Edward Frenkel] Langlands Correspondence for Loop(BookZa.org)

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CUUK1037-Frenkel March 27, 2007 20:19

Langlands Correspondence for Loop Groups

The Langlands Program was conceived initially as a bridge between Number Theory

and Automorphic Representations, and has now expanded into such areas as Geometry

and Quantum Field Theory, weaving together seemingly unrelated disciplines into a

web of tantalizing conjectures. This book provides a new chapter in this grand project.

It develops the geometric Langlands Correspondence for Loop Groups, a new

approach, from a unique perspective offered by affine Kac–Moody algebras. The

theory offers fresh insights into the world of Langlands dualities, with many

applications to Representation Theory of Infinite-dimensional Algebras, and Quantum

Field Theory. This introductory text builds the theory from scratch, with all necessary

concepts defined and the essential results proved along the way. Based on courses

taught by the author at Berkeley, the book provides many open problems which could

form the basis for future research, and is accessible to advanced undergraduate

students and beginning graduate students.

i



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ii



Langlands Correspondence

for Loop Groups

EDWARD FRENKELUniversity of California, Berkeley

iii

CAMBRIDGE UNIVERSITY PRESS

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Cambridge University PressThe Edinburgh Building, Cambridge CB2 8RU, UK

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© E. Frenkel 2007

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For my parents

Concentric geometries of transparency slightlyjoggled sink through algebras of proud

inwardlyness to collide spirally with iron arithmethics...

– E.E. Cummings, “W [ViVa]” (1931)

Contents

Preface page xi

1 Local Langlands correspondence 11.1 The classical theory 11.2 Langlands parameters over the complex field 91.3 Representations of loop groups 18

2 Vertex algebras 312.1 The center 312.2 Basics of vertex algebras 382.3 Associativity in vertex algebras 52

3 Constructing central elements 613.1 Segal–Sugawara operators 623.2 Lie algebras associated to vertex algebras 703.3 The center of a vertex algebra 763.4 Jet schemes 813.5 The center in the case of sl2 87

4 Opers and the center for a general Lie algebra 1014.1 Projective connections, revisited 1014.2 Opers for a general simple Lie algebra 1074.3 The center for an arbitrary affine Kac–Moody algebra 117

5 Free field realization 1265.1 Overview 1265.2 Finite-dimensional case 1285.3 The case of affine algebras 1355.4 Vertex algebra interpretation 1405.5 Computation of the two-cocycle 1455.6 Comparison of cohomology classes 151

6 Wakimoto modules 1616.1 Wakimoto modules of critical level 1626.2 Deforming to other levels 167

ix

x Contents

6.3 Semi-infinite parabolic induction 1786.4 Appendix: Proof of the Kac–Kazhdan conjecture 189

7 Intertwining operators 1927.1 Strategy of the proof 1927.2 The case of sl2 1977.3 Screening operators for an arbitrary g 205

8 Identification of the center with functions on opers 2148.1 Description of the center of the vertex algebra 2158.2 Identification with the algebra of functions on opers 2268.3 The center of the completed universal enveloping algebra 237

9 Structure of g-modules of critical level 2439.1 Opers with regular singularity 2449.2 Nilpotent opers 2479.3 Miura opers with regular singularities 2569.4 Categories of representations of g at the critical level 2659.5 Endomorphisms of the Verma modules 2739.6 Endomorphisms of the Weyl modules 279

10 Constructing the Langlands correspondence 29510.1 Opers vs. local systems 29710.2 Harish–Chandra categories 30110.3 The unramified case 30510.4 The tamely ramified case 32610.5 From local to global Langlands correspondence 353

Appendix 366A.1 Lie algebras 366A.2 Universal enveloping algebras 366A.3 Simple Lie algebras 368A.4 Central extensions 370A.5 Affine Kac–Moody algebras 371

References 373Index 377

Preface

The Langlands Program has emerged in recent years as a blueprint for a GrandUnified Theory of Mathematics. Conceived initially as a bridge between Num-ber Theory and Automorphic Representations, it has now expanded into suchareas as Geometry and Quantum Field Theory, weaving together seeminglyunrelated disciplines into a web of tantalizing conjectures. The Langlandscorrespondence manifests itself in a variety of ways in these diverse areasof mathematics and physics, but the same salient features, such as the ap-pearance of the Langlands dual group, are always present. This points tosomething deeply mysterious and elusive, and that is what makes this corre-spondence so fascinating.

One of the prevalent themes in the Langlands Program is the interplaybetween the local and global pictures. In the context of Number Theory, forexample, “global” refers to a number field (a finite extension of the field ofrational numbers) and its Galois group, while “local” means a local field, suchas the field of p-adic numbers, together with its Galois group. On the otherside of the Langlands correspondence we have, in the global case, automorphicrepresentations, and, in the local case, representations of a reductive group,such as GLn, over the local field.

In the geometric context the cast of characters changes: on the Galois sidewe now have vector bundles with flat connection on a complex Riemann sur-face X in the global case, and on the punctured disc D× around a point ofX in the local case. The definition of the objects on the other side of thegeometric Langlands correspondence is more subtle. It is relatively well un-derstood (after works of A. Beilinson, V. Drinfeld, G. Laumon and others)in the special case when the flat connection on our bundle has no singulari-ties. Then the corresponding objects are the so-called “Hecke eigensheaves”on the moduli spaces of vector bundles on X. These are the geometric ana-logues of unramified automorphic functions. The unramified global geometricLanglands correspondence is then supposed to assign to a flat connection onour bundle (without singularities) a Hecke eigensheaf. (This is discussed in

xi

xii Preface

a recent review [F7], among other places, where we refer the reader for moredetails.)

However, in the more general case of connections with ramification, thatis, with singularities, the geometric Langlands correspondence is much moremysterious, both in the local and in the global case. Actually, the impetusnow shifts more to the local story. This is because the flat connections thatwe consider have finitely many singular points on our Riemann surface. Theglobal ramified correspondence is largely determined by what happens on thepunctured discs around those points, which is in the realm of the local cor-respondence. So the question really becomes: what is the geometric analogueof the local Langlands correspondence?

Now, the classical local Langlands correspondence relates representations ofp-adic groups and Galois representations. In the geometric version we shouldreplace a p-adic group by the (formal) loop group G((t)), the group of mapsfrom the (formal) punctured disc D× to a complex reductive algebraic groupG. Galois representations should be replaced by vector bundles on D× witha flat connection. These are the local geometric Langlands parameters. Toeach of them we should be able to attach a representation of the formal loopgroup.

Recently, Dennis Gaitsgory and I have made a general proposal describingthese representations of loop groups. An important new element in our pro-posal is that, in contrast to the classical correspondence, the loop group nowacts on categories rather than vector spaces. Thus, the Langlands correspon-dence for loop groups is categorical: we associate categorical representationsof G((t)) to local Langlands parameters. We have proposed how to constructthese categories using representations of the affine Kac–Moody algebra g,which is a central extension of the loop Lie algebra g((t)). Therefore the lo-cal geometric Langlands correspondence appears as the result of a successfulmarriage of the Langlands philosophy and the representation theory of affineKac–Moody algebras.

Affine Kac–Moody algebras have a parameter, called the level. For a specialvalue of this parameter, called the critical level, the completed envelopingalgebra of an affine Kac–Moody algebra acquires an unusually large center.In 1991, Boris Feigin and I showed that this center is canonically isomorphicto the algebra of functions on the space of opers on D×. Opers are bundles onD× with a flat connection and an additional datum (as defined by Drinfeld–Sokolov and Beilinson–Drinfeld). Remarkably, their structure group turns outto be not G, but the Langlands dual group LG, in agreement with the generalLanglands philosophy. This is the central result, which implies that the samesalient features permeate both the representation theory of p-adic groups andthe (categorical) representation theory of loop groups.

This result had been conjectured by V. Drinfeld, and it plays an importantrole in his and A. Beilinson’s approach to the global geometric Langlands

Preface xiii

correspondence, via quantization of the Hitchin systems. The isomorphismbetween the center and functions on opers means that the category of rep-resentations of g of critical level “lives” over the space of LG-opers on D×,and the loop group G((t)) acts “fiberwise” on this category. In a nutshell, theproposal of Gaitsgory and myself is that the “fibers” of this category are thesought-after categorical representations of G((t)) corresponding to the localLanglands parameters underlying the LG-opers. This has many non-trivialconsequences, both for the local and global geometric Langlands correspon-dence, and for the representation theory of g. We hope that further study ofthese categories will give us new clues and insights into the mysteries of theLanglands correspondence.†

The goal of this book is to present a systematic and self-contained intro-duction to the local geometric Langlands correspondence for loop groups andthe related representation theory of affine Kac–Moody algebras. It covers theresearch done in this area over the last twenty years and is partially based onthe graduate courses that I have taught at UC Berkeley in 2002 and 2004. Inthe book, the entire theory is built from scratch, with all necessary conceptsdefined and all essential results proved along the way. We introduce suchconcepts as the Weil-Deligne group, Langlands dual group, affine Kac–Moodyalgebras, vertex algebras, jet schemes, opers, Miura opers, screening opera-tors, etc., and illustrate them by detailed examples. In particular, many ofthe results are first explained in the simplest case of SL2. Practically no back-ground beyond standard college algebra is required from the reader (exceptpossibly in the last chapter); we even explain some standard notions, such asuniversal enveloping algebras, in the Appendix.

In the opening chapter, we present a pedagogical overview of the classicalLanglands correspondence and a motivated step-by-step passage to the geo-metric setting. This leads us to the study of affine Kac–Moody algebras andin particular the center of the completed enveloping algebra. We then reviewin great detail the construction of a series of representations of affine Kac–Moody algebras, called Wakimoto modules. They were defined by Feigin andmyself in the late 1980s following the work of M. Wakimoto. These modulesgive us an effective tool for developing the representation theory of affine alge-bras. In particular, they are crucial in our proof of the isomorphism betweenthe spectrum of the center and opers. A detailed exposition of the Wakimotomodules and the proof of this isomorphism constitute the main part of thisbook. These results allow us to establish a deep link between the representa-tion theory of affine Kac–Moody algebras of critical level and the geometryof opers. In the closing chapter, we review the results and conjectures of

† We note that A. Beilinson has another proposal [Bei] for the local geometric Langlandscorrespondence, using representations of affine Kac–Moody algebras of integral levels lessthan critical. It would be interesting to understand the connection between his proposaland ours.

xiv Preface

Gaitsgory and myself describing the representation categories associated toopers in the framework of the Langlands correspondence. I also discuss theimplications of this for the global geometric Langlands correspondence. Theseare only the first steps of a new theory, which we hope will ultimately help usreveal the secrets of Langlands duality.

Contents

Here is a more detailed description of the contents of this book.Chapter 1 is the introduction to the subject matter of this book. We begin

by giving an overview of the local and global Langlands correspondence inthe classical setting. Since the global case is discussed in great detail inmy recent review [F7], I concentrate here mostly on the local case. Next,I explain what changes one needs to make in order to transport the localLanglands correspondence to the realm of geometry and the representationtheory of loop groups. I give a pedagogical account of Galois groups, principalbundles with connections and central extensions, among other topics. Thisdiscussion leads us to the following question: how to attach to each localgeometric Langlands parameter an abelian category equipped with an actionof the formal loop group?

In Chapter 2 we take up this question in the context of the representationtheory of affine Kac–Moody algebras. This motivates us to study the center ofthe completed enveloping algebra of g. First, we do that by elementary means,but very quickly we realize that we need a more sophisticated technique. Thistechnique is the theory of vertex algebras. We give a crash course on vertexalgebras (following [FB]), summarizing all necessary concepts and results.

Armed with these results, we begin in Chapter 3 a more in-depth studyof the center of the completed enveloping algebra of g at the critical level(we find that the center is trivial away from the critical level). We describethe center in the case of the simplest affine Kac–Moody algebra sl2 and thequasi-classical analogue of the center for an arbitrary g.

In Chapter 4 we introduce the key geometric concept of opers, introducedoriginally in [DS, BD1]. We state the main result, due to Feigin and myself[FF6, F4], that the center at the critical level, corresponding to g, is isomorphicto the algebra of functions on LG-opers.

In order to prove this result, we need to develop the theory of Wakimotomodules. This is done in Chapters 5 and 6, following [F4]. We start byexplaining the analogous theory for finite-dimensional simple Lie algebras,which serves as a prototype for our construction. Then we explain the non-trivial elements of the infinite-dimensional case, such as the cohomologicalobstruction to realizing a loop algebra in the algebra of differential operatorson a loop space. This leads to a conceptual explanation of the non-trivialityof the critical level. In Chapter 6 we complete the construction of Wakimoto

Preface xv

modules, both at the critical and non-critical levels. We prove some usefulresults on representations of affine Kac–Moody algebras, such as the Kac-Kazhdan conjecture.

Having built the theory of Wakimoto modules, we are ready to tackle theisomorphism between the center and the algebra of functions on opers. Atthe beginning of Chapter 7 we give a detailed overview of the proof of thisisomorphism. In the rest of Chapter 7 we introduce an important class ofintertwining operators between Wakimoto modules called the screening oper-ators. We use these operators and some results on associated graded algebrasin Chapter 8 to complete the proof of our main result and to identify the cen-ter with functions on opers (here we follow [F4]). In particular, we clarify theorigins of the appearance of the Langlands dual group in this isomorphism,tracing it back to a certain duality between vertex algebras known as the W-algebras. At the end of the chapter we discuss the vertex Poisson structureon the center and identify the action of the center on Wakimoto modules withthe Miura transformation.

In Chapter 9 we undertake a more in-depth study of representations of affineKac–Moody algebras of critical level. We first introduce (following [BD1] and[FG2]) certain subspaces of the space of opers on D×: opers with regularsingularities and nilpotent opers, and explain the interrelations between them.We then discuss Miura opers with regular singularities and the action of theMiura transformation on them, following [FG2]. Finally, we describe theresults of [FG2] and [FG6] on the algebras of endomorphisms of the Vermamodules and the Weyl modules of critical level.

In Chapter 10 we bring together the results of this book to explain theproposal for the local geometric Langlands correspondence made by Gaitsgoryand myself. We review the results and conjectures of our works [FG1]–[FG6],emphasizing the analogies between the geometric and the classical Langlandscorrespondence. We discuss in detail the interplay between opers and localsystems. We then consider the simplest local system; namely, the trivial one.The corresponding categorical representations of G((t)) are the analogues ofunramified representations of p-adic groups. Already in this case we willsee rather non-trivial elements, which emulate the corresponding elements ofthe classical theory and at the same time generalize them in a non-trivialway. The next, and considerably more complicated, example is that of localsystems on D× with regular singularity and unipotent monodromy. These arethe analogues of the tamely ramified representations of the Galois group ofa p-adic field. The corresponding categories turn out to be closely related tocategories of quasicoherent sheaves on the Springer fibers, which are algebraicsubvarieties of the flag variety of the Langlands dual group. We summarizethe conjectures and results of [FG2] concerning these categories and illustratethem by explicit computations in the case of sl2. We also formulate some

xvi Preface

open problems in this direction. Finally, we discuss the implications of thisapproach for the global Langlands correspondence.

Acknowledgments

I thank Boris Feigin and Dennis Gaitsgory for their collaboration on our jointworks reviewed in this book.

I have benefited over the years from numerous conversations with AlexanderBeilinson, David Ben-Zvi, Joseph Bernstein, Vladimir Drinfeld, Victor Kacand David Kazhdan on the Langlands correspondence and representations ofaffine Kac–Moody algebras. I am grateful to all of them. I also thank R.Bezrukavnikov and V. Ginzburg for their useful comments on some questionsrelated to the material of the last chapter.

I thank Alex Barnard for letting me use his notes of my graduate courseat UC Berkeley in preparation of Chapters 2 and 3. I am also indebted toArthur Greenspoon, Reimundo Heluani, Valerio Toledano Laredo and XinwenZhu for their valuable comments on the draft of the book. I am grateful toDavid Tranah of Cambridge University Press for his expert guidance andfriendly advice throughout the publication process.

In the course of writing this book I received generous support from DARPAthrough its Program “Fundamental Advances in Theoretical Mathematics”. Iwas also partially supported by the National Science Foundation through thegrant DMS-0303529. I gratefully acknowledge this support.

1

Local Langlands correspondence

In this introductory chapter we explain in detail what we mean by “local Lang-lands correspondence for loop groups.” We begin by giving a brief overviewof the local Langlands correspondence for reductive groups over local non-archimedian fields, such as Fq((t)), the field of Laurent power series over afinite field Fq. We wish to understand an analogue of this correspondencewhen Fq is replaced by the field C of complex numbers. The role of the reduc-tive group will then be played by the formal loop group G(C((t))). We discuss,following [FG2], how the setup of the Langlands correspondence should changein this case. This discussion will naturally lead us to categories of represen-tations of the corresponding affine Kac–Moody algebra g equipped with anaction of G(C((t))), the subject that we will pursue in the rest of this book.

1.1 The classical theory

The local Langlands correspondence relates smooth representations of reduc-tive algebraic groups over local fields and representations of the Galois groupof this field. We start out by defining these objects and explaining the mainfeatures of this correspondence. As the material of this section serves moti-vational purposes, we will only mention those aspects of this story that aremost relevant for us. For a more detailed treatment, we refer the reader tothe informative surveys [V, Ku] and references therein.

1.1.1 Local non-archimedian fields

Let F be a local non-archimedian field. In other words, F is the field Qp ofp-adic numbers, or a finite extension of Qp, or F is the field Fq((t)) of formalLaurent power series with coefficients in Fq, the finite field with q elements.

We recall that for any q of the form pn, where p is a prime, there is a unique,up to isomorphism, finite field of characteristic p with q elements. An element

1

2 Local Langlands correspondence

of Fq((t)) is an expression of the form∑n∈Z

antn, an ∈ Fq,

such that an = 0 for all n less than some integer N . In other words, theseare power series infinite in the positive direction and finite in the negativedirection. Recall that a p-adic number may also be represented by a series∑

n∈Zbnp

n, bn ∈ 0, 1, . . . , p− 1,

such that bn = 0 for all n less than some integer N . We see that elements ofQp look similar to elements of Fp((t)). Both fields are complete with respectto the topology defined by the norm taking value α−N on the above series ifaN 6= 0 and an = 0 for all n < N , where α is a fixed positive real numberbetween 0 and 1. But the laws of addition and multiplication in the two fieldsare different: with “carry” to the next digit in the case of Qp, but without“carry” in the case of Fp((t)). In particular, Qp has characteristic 0, whileFq((t)) has characteristic p. More generally, elements of a finite extension ofQp look similar to elements of Fq((t)) for some q = pn, but, again, the rules ofaddition and multiplication, as well as their characteristics, are different.

1.1.2 Smooth representations of GLn(F )

Now consider the group GLn(F ), where F is a local non-archimedian field. Arepresentation of GLn(F ) on a complex vector space V is a homomorphismπ : GLn(F ) → EndV such that π(gh) = π(g)π(h) and π(1) = Id. Define atopology on GLn(F ) by stipulating that the base of open neighborhoods of1 ∈ GLn(F ) is formed by the congruence subgroups KN , N ∈ Z+. In the casewhen F = Fq((t)), the group KN is defined as follows:

KN = g ∈ GLn(Fq[[t]]) | g ≡ 1 mod tN,

and for F = Qp it is defined in a similar way. For each v ∈ V we obtain a mapπ(·)v : GLn(F ) → V, g 7→ π(g)v. A representation (V, π) is called smooth ifthe map π(·)v is continuous for each v, where we give V the discrete topology.In other words, V is smooth if for any vector v ∈ V there exists N ∈ Z+ suchthat

π(g)v = v, ∀g ∈ KN .

We are interested in describing the equivalence classes of irreducible smoothrepresentations of GLn(F ). Surprisingly, those turn out to be related toobjects of a different kind: n-dimensional representations of the Galois groupof F .

1.1 The classical theory 3

1.1.3 The Galois group

Suppose that F is a subfield ofK. Then the Galois group Gal(K/F ) consistsof all automorphisms σ of the field K such that σ(y) = y for all y ∈ F .

Let F be a field. The algebraic closure of F is a field obtained by adjoiningto F the roots of all polynomials with coefficients in F . In the case whenF = Fq((t)) some of the extensions of F may be non-separable. An example ofsuch an extension is the field Fq((t1/p)). The polynomial defining this extensionis xp − t, but in Fq((t1/p)) it has multiple roots because

xp − (t1/p)p = (x− t1/p)p.

The Galois group Gal(Fq((t1/p)),Fq((t))) of this extension is trivial, even thoughthe degree of the extension is p.

This extension should be contrasted to the separable extensions Fq((t1/n)),where n is not divisible by p. This extension is defined by the polynomialxn − t, which now has no multiple roots:

xn − t =n−1∏i=0

(x− ζit1/n),

where ζ is a primitive nth root of unity in the algebraic closure of Fq. Thecorresponding Galois group is identified with the group (Z/nZ)×, the groupof invertible elements of Z/nZ.

We wish to avoid the non-separable extensions, because they do not con-tribute to the Galois group. (There are no non-separable extensions if F hascharacteristic zero, e.g., for F = Qp.) Let F be the maximal separable ex-tension inside a given algebraic closure of F . It is uniquely defined up toisomorphism.

Let Gal(F/F ) be the absolute Galois group of F . Its elements are theautomorphisms σ of the field F such that σ(y) = y for all y ∈ F .

To gain some experience with Galois groups, let us look at the Galois groupGal(Fq/Fq). Here Fq is the algebraic closure of Fq, which can be defined asthe inductive limit of the fields FqN , N ∈ Z+, with respect to the naturalembeddings FqN → FqM for N dividing M . Therefore Gal(Fq/Fq) is isomor-phic to the inverse limit of the Galois groups Gal(FqN /Fq) with respect to thenatural surjections

Gal(FqM /Fq) Gal(FqN /Fq), ∀N |M.

The group Gal(FqN /Fq) is easy to describe: it is generated by the Frobeniusautomorphism x 7→ xq (note that it stabilizes Fq), which has order N , sothat Gal(FqN /Fq) ' Z/NZ. Therefore we find that

Gal(Fq/Fq) ' Z def= lim←−

Z/NZ,


where we have the surjective maps Z/MZ Z/NZ for N |M . The group Zcontains Z as a subgroup.

Let F = Fq((t)). Observe that we have a natural map Gal(F/F ) →Gal(Fq/Fq) obtained by applying an automorphism of F to Fq ⊂ F . A similarmap also exists when F has characteristic 0. Let WF be the preimage of thesubgroup Z ⊂ Gal(Fq/Fq). This is the Weil group of F . Let ν be the cor-responding homomorphism WF → Z. Let W ′F = WF n C be the semi-directproduct of WF and the one-dimensional complex additive group C, where WF

acts on C by the formula

σxσ−1 = qν(σ)x, σ ∈WF , x ∈ C. (1.1.1)

This is the Weil–Deligne group of F .An n-dimensional complex representation of W ′F is by definition a homo-

morphism ρ′ : W ′F → GLn(C), which may be described as a pair (ρ, u),where ρ is an n-dimensional representation of WF , u ∈ gln(C), and we haveρ(σ)uρ(σ)−1 = qν(σ)u for all σ ∈ WF . The group WF is topological, with re-spect to the Krull topology (in which the open neighborhoods of the identityare the normal subgroups of finite index). The representation (ρ, u) is calledadmissible if ρ is continuous (equivalently, factors through a finite quotientof WF ) and semisimple, and u is a nilpotent element of gln(C).

The group W ′F was introduced by P. Deligne [De2]. The idea is that byadjoining the nilpotent element u to WF we obtain a group whose complexadmissible representations are the same as continuous `-adic representationsof WF (where ` 6= p is a prime).

1.1.4 The local Langlands correspondence for GLn

Now we are ready to state the local Langlands correspondence for the groupGLn over a local non-archimedian field F . It is a bijection between two differ-ent sorts of data. One is the set of equivalence classes of irreducible smoothrepresentations of GLn(F ). The other is the set of equivalence classes of n-dimensional admissible representations of W ′F . We represent it schematicallyas follows:

n-dimensional admissiblerepresentations of W ′F

⇐⇒ irreducible smoothrepresentations of GLn(F )

This correspondence is supposed to satisfy an overdetermined system ofconstraints which we will not recall here (see, e.g., [Ku]).

The local Langlands correspondence for GLn is a theorem. In the casewhen F = Fq((t)) it has been proved in [LRS], and when F = Qp or its finiteextension in [HT] and also in [He]. We refer the readers to these papers andto the review [C] for more details.


Despite an enormous effort made in the last two decades to understandit, the local Langlands correspondence still remains a mystery. We do knowthat the above bijection exists, but we cannot yet explain in a completelysatisfactory way why it exists. We do not know the deep underlying reasonsthat make such a correspondence possible. One way to try to understand itis to see how general it is. In the next section we will discuss one possiblegeneralization of this correspondence, where we replace the group GLn by anarbitrary reductive algebraic group defined over F .

1.1.5 Generalization to other reductive groups

Let us replace the group GLn by an arbitrary connected reductive group G

over a local non-archimedian field F . The group G(F ) is also a topologicalgroup, and there is a notion of smooth representation of G(F ) on a complexvector space. It is natural to ask whether we can relate irreducible smoothrepresentations of G(F ) to representations of the Weil–Deligne group W ′F .This question is addressed in the general local Langlands conjectures. Itwould take us too far afield to try to give here a precise formulation of theseconjectures. So we will only indicate some of the objects involved, referringthe reader to the articles [V, Ku], where these conjectures are described ingreat detail.

Recall that in the case when G = GLn the irreducible smooth represen-tations are parametrized by admissible homomorphisms W ′F → GLn(C). Inthe case of a general reductive group G, the representations are conjecturallyparametrized by admissible homomorphisms from W ′F to the so-called Lang-lands dual group LG, which is defined over C.

In order to explain the notion of the Langlands dual group, consider firstthe group G over the closure F of the field F . All maximal tori T of thisgroup are conjugate to each other and are necessarily split, i.e., we have anisomorphism T (F ) ' (F

×)n. For example, in the case of GLn, all maximal

tori are conjugate to the subgroup of diagonal matrices. We associate to T (F )two lattices: the weight lattice X∗(T ) of homomorphisms T (F ) → F

×and

the coweight lattice X∗(T ) of homomorphisms F× → T (F ). They contain

the sets of roots ∆ ⊂ X∗(T ) and coroots ∆∨ ⊂ X∗(T ), respectively. Thequadruple (X∗(T ), X∗(T ),∆,∆∨) is called the root data for G over F . Theroot data determines G up to an isomorphism defined over F . The choice ofa Borel subgroup B(F ) containing T (F ) is equivalent to a choice of a basisin ∆; namely, the set of simple roots ∆s, and the corresponding basis ∆∨s in∆∨.

Now given γ ∈ Gal(F/F ), there is g ∈ G(F ) such that g(γ(T (F ))g−1 =T (F ) and g(γ(B(F ))g−1 = B(F ). Then g gives rise to an automorphism of


the based root data (X∗(T ), X∗(T ),∆s,∆∨s ). Thus, we obtain an action ofGal(F/F ) on the based root data.

Let us now exchange the lattices of weights and coweights and the sets ofsimple roots and coroots. Then we obtain the based root data

(X∗(T ), X∗(T ),∆∨s ,∆s)

of a reductive algebraic group over C, which is denoted by LG. For instance,the group GLn is self-dual, the dual of SO2n+1 is Sp2n, the dual of Sp2n isSO2n+1, and SO2n is self-dual.

The action of Gal(F/F ) on the based root data gives rise to its action onLG. The semi-direct product LG = Gal(F/F )nLG is called the Langlandsdual group of G.

The local Langlands correspondence for the group G(F ) relates the equiva-lence classes of irreducible smooth representations of G(F ) to the equivalenceclasses of admissible homomorphisms W ′F → LG. However, in general thiscorrespondence is much more subtle than in the case of GLn. In particular,we need to consider simultaneously representations of all inner forms of G,and a homomorphism W ′F → LG corresponds in general not to a single ir-reducible representation of G(F ), but to a finite set of representations calledan L-packet. To distinguish between them, we need additional data (see [V]for more details; some examples are presented in Section 10.4.1 below). Butin the first approximation we can say that the essence of the local Langlandscorrespondence is that

irreducible smooth representations of G(F ) are parameterizedin terms of admissible homomorphisms W ′F → LG.

1.1.6 On the global Langlands correspondence

We close this section with a brief discussion of the global Langlands corre-spondence and its connection to the local one. We will return to this subjectin Section 10.5.

Let X be a smooth projective curve over Fq. Denote by F the field Fq(X)of rational functions on X. For any closed point x of X we denote by Fx

the completion of F at x and by Ox its ring of integers. If we choose a localcoordinate tx at x (i.e., a rational function on X which vanishes at x to orderone), then we obtain isomorphisms Fx ' Fqx((tx)) and Ox ' Fqx [[tx]], whereFqx

is the residue field of x; in general, it is a finite extension of Fq containingqx = qdeg(x) elements.

Thus, we now have a local field attached to each point of X. The ringA = AF of adeles of F is by definition the restricted product of the fields Fx,where x runs over the set |X| of all closed points of X. The word “restricted”means that we consider only the collections (fx)x∈|X| of elements of Fx in


which fx ∈ Ox for all but finitely many x. The ring A contains the field F ,which is embedded into A diagonally, by taking the expansions of rationalfunctions on X at all points.

While in the local Langlands correspondence we considered irreduciblesmooth representations of the group GLn over a local field, in the globalLanglands correspondence we consider irreducible automorphic represen-tations of the group GLn(A). The word “automorphic” means, roughly, thatthe representation may be realized in a reasonable space of functions on thequotient GLn(F )\GLn(A) (on which the group GLn(A) acts from the right).

On the other side of the correspondence we consider n-dimensional repre-sentations of the Galois group Gal(F/F ), or, more precisely, the Weil groupWF , which is a subgroup of Gal(F/F ) defined in the same way as in the localcase.

Roughly speaking, the global Langlands correspondence is a bijection be-tween the set of equivalence classes of n-dimensional representations of WF

and the set of equivalence classes of irreducible automorphic representationsof GLn(A):

n-dimensional representationsof WF

⇐⇒ irreducible automorphicrepresentations of GLn(A)

The precise statement is more subtle. For example, we should considerthe so-called `-adic representations of the Weil group (while in the local casewe considered the admissible complex representations of the Weil–Delignegroup; the reason is that in the local case those are equivalent to the `-adicrepresentations). Moreover, under this correspondence important invariantsattached to the objects appearing on both sides (Frobenius eigenvalues onthe Galois side and the Hecke eigenvalues on the other side) are supposed tomatch. We refer the reader to Part I of the review [F7] for more details.

The global Langlands correspondence has been proved for GL2 in the 1980’sby V. Drinfeld [Dr1]–[Dr4] and more recently by L. Lafforgue [Laf] for GLn

with an arbitrary n.The global and local correspondences are compatible in the following sense.

We can embed the Weil group WFxof each of the local fields Fx into the global

Weil group WF . Such an embedding is not unique, but it is well-defined upto conjugation in WF . Therefore an equivalence class σ of n-dimensionalrepresentations of WF gives rise to a well-defined equivalence class σx of n-dimensional representations of WFx

for each x ∈ X. By the local Langlandscorrespondence, to σx we can attach an equivalence class of irreducible smoothrepresentations of GLn(Fx). Choose a representation πx in this equivalenceclass. Then the automorphic representation of GLn(A) corresponding to σ


is isomorphic to the restricted tensor product⊗′

x∈X πx. This is a very non-trivial statement, because a priori it is not clear why this tensor product maybe realized in the space of functions on the quotient GLn(F )\GLn(A).

As in the local story, we may also wish to replace the group GLn by an arbi-trary reductive algebraic group defined over F . The general global Langlandsconjecture predicts, roughly speaking, that irreducible automorphic represen-tations of G(A) are related to homomorphisms WF → LG. But, as in thelocal case, the precise formulation of the conjecture for a general reductivegroup is much more intricate (see [Art]).

Finally, the global Langlands conjectures can also be stated over numberfields (where they in fact originated). Then we take as the field F a finiteextension of the field Q of rational numbers. Consider for example the case ofQ itself. It is known that the completions of Q are (up to isomorphism) thefields of p-adic numbers Qp for all primes p (non-archimedian) and the field Rof real numbers (archimedian). So the primes play the role of points of an al-gebraic curve over a finite field (and the archimedian completion correspondsto an infinite point, in some sense). The ring of adeles AQ is defined in thesame way as in the function field case, and so we can define the notion of anautomorphic representation of GLn(AQ) or a more general reductive group.Conjecturally, to each equivalence class of n-dimensional representations ofthe Galois group Gal(Q/Q) we can attach an equivalence class of irreducibleautomorphic representations of GLn(AQ), but this correspondence is not ex-pected to be a bijection because in the number field case it is known thatsome of the automorphic representations do not correspond to any Galoisrepresentations.

The Langlands conjectures in the number field case lead to very importantand unexpected results. Indeed, many interesting representations of Galoisgroups can be found in “nature”. For example, the group Gal(Q/Q) will acton the geometric invariants (such as the etale cohomologies) of an algebraicvariety defined over Q. Thus, if we take an elliptic curve E over Q, then wewill obtain a two-dimensional Galois representation on its first etale cohomol-ogy. This representation contains a lot of important information about thecurve E, such as the number of points of E over Z/pZ for various primes p.The Langlands correspondence is supposed to relate these Galois representa-tions to automorphic representations of GL2(AF ) in such a way that the dataon the Galois side, like the number of points of E(Z/pZ), are translated intosomething more tractable on the automorphic side, such as the coefficientsin the q-expansion of the modular forms that encapsulate automorphic rep-resentations of GL2(AQ). This leads to some startling consequences, such asthe Taniyama-Shimura conjecture. For more on this, see [F7] and referencestherein.

The Langlands correspondence has proved to be easier to study in the func-tion field case. The main reason is that in the function field case we can use

1.2 Langlands parameters over the complex field 9

the geometry of the underlying curve and various moduli spaces associated tothis curve. A curve can also be considered over the field of complex numbers.Some recent results show that a version of the global Langlands correspon-dence also exists for such curves. The local counterpart of this correspondenceis the subject of this book.

1.2 Langlands parameters over the complex field

We now wish to find a generalization of the local Langlands conjectures inwhich we replace the field F = Fq((t)) by the field C((t)). We would like tosee how the ideas and patterns of the Langlands correspondence play out inthis new context, with the hope of better understanding the deep underlyingstructures behind this correspondence.

So from now on G will be a connected reductive group over C, and G(F )the group G((t)) = G(C((t))), also known as the loop group; more precisely,the formal loop group, with the word “formal” referring to the algebra offormal Laurent power series C((t)) (as opposed to the groupG(C[t, t−1]), whereC[t, t−1] is the algebra of Laurent polynomials, which may be viewed as thegroup of maps from the unit circle |t| = 1 to G, or “loops” in G).

Thus, we wish to study smooth representations of the loop group G((t))and try to relate them to some “Langlands parameters,” which we expect, byanalogy with the case of local non-archimedian fields described above, to berelated to the Galois group of C((t)) and the Langlands dual group LG.

The local Langlands correspondence for loop groups that we discuss in thisbook may be viewed as the first step in carrying the ideas of the LanglandsProgram to the realm of complex algebraic geometry. In particular, it hasfar-reaching consequences for the global geometric Langlands correspondence(see Section 10.5 and [F7] for more details). This was in fact one of themotivations for this project.

1.2.1 The Galois group and the fundamental group

We start by describing the Galois group Gal(F/F ) for F = C((t)). Observethat the algebraic closure F of F is isomorphic to the inductive limit of thefields C((t1/n)), n ≥ 0, with respect to the natural inclusions C((t1/n)) →C((t1/m)) for n dividing m. Hence Gal(F/F ) is the inverse limit of the Galoisgroups

Gal(C((t1/n))/C((t))) ' Z/nZ,

where k ∈ Z/nZ corresponds to the automorphism of C((t1/n)) sending t1/n

to e2πik/nt1/n. The result is that

Gal(F/F ) ' Z,


where Z is the profinite completion of Z that we have encountered before inSection 1.1.3.

Note, however, that in our study of the Galois group of Fq((t)) the groupZ appeared as its quotient corresponding to the Galois group of the field ofcoefficients Fq. Now the field of coefficients is C, hence algebraically closed,and Z is the entire Galois group of C((t)).

The naive analogue of the Langlands parameter would be an equivalenceclass of homomorphisms Gal(F/F ) → LG, i.e., a homomorphism Z → LG.Since G is defined over C and hence all of its maximal tori are split, the groupG((t)) also contains a split torus T ((t)), where T is a maximal torus of G (butit also contains non-split maximal tori, as the field C((t)) is not algebraicallyclosed). Therefore the Langlands dual group LG is the direct product of theGalois group and the group LG. Because it is a direct product, we may, andwill, restrict our attention to LG. In order to simplify our notation, fromnow on we will denote LG simply by LG.

A homomorphism Z → LG necessarily factors through a finite quotientZ→ Z/nZ. Therefore the equivalence classes of homomorphisms Z→ LG arethe same as the conjugacy classes of LG of finite order. There are too few ofthese to have a meaningful generalization of the Langlands correspondence.Therefore we look for a more sensible alternative.

Let us recall the connection between Galois groups and fundamental groups.Let X be an algebraic variety over C. If Y → X is a covering of X, then thefield C(Y ) of rational functions on Y is an extension of the field F = C(X)of rational functions on X. The deck transformations of the cover, i.e., auto-morphisms of Y which induce the identity on X, give rise to automorphismsof the field C(Y ) preserving C(X) ⊂ C(Y ). Hence we identify the Galoisgroup Gal(C(Y )/C(X)) with the group of deck transformations. If our coveris unramified, then this group may be identified with a quotient of the funda-mental group of X. Otherwise, this group is isomorphic to a quotient of thefundamental group of X with the ramification divisor thrown out.

In particular, we obtain that the Galois group of the maximal unramifiedextension of C(X) (which we can view as the field of functions of the “maximalunramified cover” of X) is the profinite completion of the fundamental groupπ1(X) of X. Likewise, for any divisor D ⊂ X the Galois group of the maximalextension of C(X) unramified away from D is the profinite completion ofπ1(X\D). We denote it by πalg

1 (X\D). The algebraic closure of C(X) is theinductive limit of the fields of functions on the maximal covers of X ramifiedat various divisors D ⊂ X with respect to natural inclusions corresponding tothe inclusions of the divisors. Hence the Galois group Gal(C(X)/C(X)) is theinverse limit of the groups πalg

1 (X\D) with respect to the maps πalg1 (X\D′)→

πalg1 (X\D) for D ⊂ D′.Strictly speaking, in order to define the fundamental group of X we need

to pick a reference point x ∈ X. In the above discussion we have tacitly


picked as the reference point the generic point of X, i.e., Spec C(X) → X.But for different choices of x the corresponding fundamental groups will beisomorphic anyway, and so the equivalences classes of their representations(which is what we are after) will be the same. Therefore we will ignore thechoice of a reference point.

Going back to our setting, we see that Gal(F/F ) for F = C((t)) is indeedZ, the profinite completion of the “topological” fundamental group of thepunctured disc D×. Thus, the naive Langlands parameters correspond to ho-momorphisms πalg

1 (D×)→ LG. But, in the complex setting, homomorphismsπ1(X)→ LG can be obtained from more geometrically refined data; namely,from bundles with flat connections, which we will discuss presently.

1.2.2 Flat bundles

Suppose that X is a real manifold and E is a complex vector bundle over X.Thus, we are given a map p : E → X, satisfying the following condition: eachpoint x ∈ X has a neighborhood U such that the preimage of U in E underp is isomorphic to U × Cn. However, the choice of such a trivialization of Eover U is not unique. Two different trivializations differ by a smooth mapU → GLn(C). We can describe E more concretely by choosing a cover of Xby open subsets Uα, α ∈ A, and picking trivializations of E on each Uα. Thenthe difference between the two induced trivializations on the overlap Uα ∩Uβ

will be accounted for by a function gαβ : Uα ∩ Uβ → GLn(C), which is calledthe transition function. The transition functions satisfy the importanttransitivity property: on triple overlaps Uα ∩Uβ ∩Uγ we have gαγ = gαβgβγ .

This describes the data of an ordinary vector bundle over X. The data ofa flat vector bundle E on X are a transitive system of preferred trivial-izations of E over a sufficiently small open subset of any point of X, with thedifference between two preferred trivializations given by a constant GLn(C)-valued function. Thus, concretely, a flat bundle may be described by choosinga sufficiently fine cover of X by open subsets Uα, α ∈ A, and picking trivial-izations of E on each Uα that belong to our preferred system of trivializations.Then the transition functions gαβ : Uα ∩ Uβ → GLn(C) are constant func-tions, which satisfy the transitivity property. This should be contrasted to thecase of ordinary bundles, for which the transition functions may be arbitrarysmooth GLn(C)-valued functions.

From the perspective of differential geometry, the data of a preferred systemof trivializations are neatly expressed by the data of a flat connection.These are precisely the data necessary to differentiate the sections of E.

More precisely, let OX be the sheaf of smooth functions on X and T thesheaf of smooth vector fields on X. Denote by End(E) the bundle of endo-morphisms of E. By a slight abuse of notation we will denote by the samesymbol End(E) the sheaf of its smooth sections. A connection on E is a


map

∇ : T −→ End(E),

which takes a vector field ξ ∈ T (U) defined on an open subset U ⊂ X to anendomorphism ∇ξ of E over U . It should have the following properties:

• It is OX linear: ∇ξ+η = ∇ξ +∇η and ∇fξ = f∇ξ for f ∈ OX .• It satisfies the Leibniz rule: ∇ξ(fφ) = f∇ξ(φ) + φ · ξ(f) for f ∈ OX andφ ∈ End(E).

A connection is called flat if it has the additional property that ∇ is a Liealgebra homomorphism

[∇ξ,∇χ] = ∇[ξ,χ], (1.2.1)

where we consider the natural Lie algebra structures on T and End(E).Let us discuss a more concrete realization of connections. On a small enough

open subset U we can pick coordinates xi, i = 1, . . . , N , and trivialize thebundle E. Then by OX -linearity, to define the restriction of a connection∇ to U it is sufficient to write down the operators ∇∂xi

, where we use the

notation ∂xi =∂

∂xi. The Leibniz rule implies that these operators must have

the form

∇∂xi= ∂xi +Ai(x),

where Ai is an n× n matrix-valued smooth function on U .The flatness condition then takes the form

[∇∂xi,∇∂xj

] = ∂xiAj − ∂xjAi + [Ai, Aj ] = 0. (1.2.2)

So, for a very concrete description of a flat connection in the real case we cansimply specify matrices Ai locally, satisfying the above equations. But weshould also make sure that they transform in the correct way under changesof coordinates and changes of trivialization of the bundle E. If we introducenew coordinates such that xi = xi(y), then we find that

∂yi =n∑

j=1

∂xj

∂yi∂xj

,

and so

∇∂yj=

n∑j=1

∂xj

∂yi∇∂xj

.

A given section of E that appears as a Cn-valued function f(x) on U withrespect to the old trivialization will appear as the function g(x) · f(x) withrespect to a new trivialization, where g(x) is the GLn(C)-valued transitionfunction. Therefore the connection operators will become

g∇∂xig−1 = ∇∂xi

+ gAig−1 − (∂xig)g

−1.


Using the connection operators, we construct a transitive system of pre-ferred local trivializations of our bundle as follows: locally, on a small opensubset U , consider the system of the first order differential equations

∇ξΦ(x) = 0, ξ ∈ T (U). (1.2.3)

For this system to make sense the flatness condition (1.2.2) should be satisfied.(Otherwise, the right hand side of (1.2.2), which is a matrix-valued function,would have to annihilate Φ(x). Therefore we would have to restrict Φ(x) tothe kernel of this matrix-valued function.) In this case, the standard theoremsof existence and uniqueness of solutions of linear differential equations tell usthat for each point x0 ∈ U and each vector v in the fiber Ex0 of E at x0 ithas a unique local solution Φv satisfying the initial condition Φ(x0) = v. Thesolutions Φv for different v ∈ Ex0 are sections of E over U , which are calledhorizontal. They give us a transitive system of preferred identifications ofthe fibers of E over the points of U : Ex ' Ey,x,y ∈ U . If we identify one ofthese fibers, say the one at x0, with Cn, then we obtain a trivialization of Eover U . Changing the identification Ex0 ' Cn would change this trivializationonly by a constant function U → GLn. Thus, we obtain the desired preferredsystem of trivializations of E.

Put differently, the data of a flat connection give us a preferred system ofidentifications of nearby fibers of E, which are locally transitive. But if weuse them to identify the fibers of E lying over a given path in X startingand ending at a point x0, then we may obtain a non-trivial automorphismof the fiber Ex0 , the monodromy of the flat connection along this path.(More concretely, this monodromy may be found by considering solutions ofthe system (1.2.3) defined in a small tubular neighborhood of our path.) Thisautomorphism will depend only on the homotopy class of our path, and soany flat connection on E gives rise to a homomorphism π1(X,x0)→ AutEx0 ,where AutEx0 is the group of automorphisms of Ex0 . If we identify Ex0 withCn, we obtain a homomorphism π1(X,x0)→ GLn(C).

Thus, we assign to a flat bundle on X of rank n an equivalence class ofhomomorphisms π1(X,x0)→ GLn(C).

In the case when X is the punctured disc D×, or a smooth projective com-plex curve, equivalence classes of homomorphisms π1(X,x0) → GLn(C) areprecisely what we wish to consider as candidates for the Langlands parametersin the complex setting. That is why we are interested in their reformulationsas bundles with flat connections.

1.2.3 Flat bundles in the holomorphic setting

Suppose now that X is a complex algebraic variety and E is a holomorphicvector bundle over X. Recall that giving a vector bundle E over X the


structure of a holomorphic vector bundle is equivalent to specifying which sec-tions of E are considered to be holomorphic. On an open subset U of X withholomorphic coordinates zi, i = 1, . . . , N , and anti-holomorphic coordinateszi, i = 1, . . . , N , specifying the holomorphic sections is equivalent to definingthe ∂-operators ∇∂zi

, i = 1, . . . , N , on the sections of E. The holomorphicsections of E are then precisely those annihilated by these operators. Thus,we obtain an action of the anti-holomorphic vector fields on all smooth sec-tions of E. This action of anti-holomorphic vector fields gives us “half” of thedata of a flat connection. So, to specify a flat connection on a holomorphicvector bundle E we only need to define an action of the (local) holomorphicvector fields on X on the holomorphic sections of E.

On a sufficiently small open subset U of X we may now choose a trivializa-tion compatible with the holomorphic structure, so that we have ∇∂zi

= ∂zi.

Then extending these data to the data of a flat connection on E amounts toconstructing operators

∇∂zi= ∂zi

+Ai(z),

where the flatness condition demands that the Ai’s be holomorphic matrixvalued functions of zj , j = 1, . . . , N , such that

[∇∂zi,∇∂zj

] = ∂ziAj − ∂zjAi + [Ai, Aj ] = 0.

In particular, if X is a complex curve, i.e., N = 1, any holomorphic connectionon a holomorphic vector bundle gives rise to a flat connection, and hence anequivalence class of homomorphisms π1(X)→ GLn(C).†

This completes the story of flat connections on vector bundles, or equiva-lently, principal GLn-bundles. Next, we define flat connections on principalG-bundles, where G is a complex algebraic group.

1.2.4 Flat G-bundles

Recall that a principal G-bundle over a manifold X is a manifold P whichis fibered over X such that there is a natural fiberwise right action of G onP, which is simply-transitive along each fiber. In addition, it is locally trivial:each point has a sufficient small open neighborhood U ⊂ X over which thebundle P may be trivialized, i.e., there is an isomorphism tU : P|U

∼−→ U ×Gcommuting with the right actions of G. More concretely, we may describe Pby choosing a cover ofX by open subsets Uα, α ∈ A, and picking trivializationsof P on each Uα. Then the difference between the two induced trivializationson the overlap Uα ∩ Uβ will be described by the transition functions gαβ :Uα ∩Uβ → G, which satisfy the transitivity condition, as in the case of GLn.

† Note, however, that some holomorphic vector bundles on curves do not carry any holo-morphic connections; for instance, a line bundle carries a holomorphic connection if andonly if it has degree 0.


A flat structure on such a bundle P is a preferred system of locally tran-sitive identifications of the fibers of P, or equivalently, a preferred system oftransitive local trivializations (this means that the transition functions gαβ areconstant G-valued functions). In order to give a differential geometric realiza-tion of flat connections, we use the Tannakian formalism. It says, roughly, thatG may be reconstructed from the category of its finite-dimensional represen-tations equipped with the structure of the tensor product, satisfying variouscompatibilities, and the “fiber functor” to the category of vector spaces.

Given a principal G-bundle P on X and a finite-dimensional representationV of G we can form the associated vector bundle

VP = P ×GV.

Thus we obtain a functor from the category of finite-dimensional representa-tions of G to the category of vector bundles on X. Both of these categoriesare tensor categories and it is clear that the above functor is actually a tensorfunctor. The Tannakian formalism allows us to reconstruct the principal G-bundle P from this functor. In other words, the data of a principal G-bundleon X are encoded by the data of a collection of vector bundles on X labeledby finite-dimensional representations of G together with the isomorphisms(V ⊗ W )P ' VR ⊗ WP for each pair of representations V,W and variouscompatibilities stemming from the tensor properties of this functor.

In order to define a flat connection on a holomorphic principal G-bundleP we now need to define a flat connection on each of the associated vectorbundles VP in a compatible way. What this means is that on each sufficientlysmall open subset U ⊂ X, after choosing a system of local holomorphic coor-dinates zi, i = 1, . . . , N , on U and a holomorphic trivialization of P over U ,the data of a flat connection on P are given by differential operators

∇∂zi= ∂zi

+Ai(z),

where now the Ai’s are holomorphic functions on U with values in the Liealgebra g of G.

The transformation properties of the operators ∇∂ziunder changes of co-

ordinates and trivializations are given by the same formulas as above. Inparticular, under a change of trivialization given by a holomorphic G-valuedfunction g on U we have

∂zi +Ai 7→ ∂zi + gAig−1 − (∂zig)g

−1. (1.2.4)

Such transformations are called the gauge transformations.The meaning of this formula is as follows: the expression gAig

−1−(∂zig)g−1

appearing in the right hand side is a well-defined element of EndV for anyfinite-dimensional representation V of G. The Tannakian formalism discussedabove then implies that there is a “universal” element of g whose action onV is given by this formula.


1.2.5 Regular vs. irregular singularities

In the same way as in the case of GLn, we obtain that a holomorphic principalG-bundle on a complex variety X with a holomorphic flat connection givesrise to an equivalence class of homomorphisms from the fundamental groupof X to G. But does this set up a bijection of the corresponding equivalenceclasses? If X is compact, this is indeed the case, but, if X is not compact,then there are more flat bundles than there are representations of π1(X). Inorder to obtain a bijection, we need to impose an additional condition on theconnection; we need to require that it has regular singularities at infinity.

If X is a curve obtained from a projective (hence compact) curve X byremoving finitely many points, this condition means that the connection op-erator ∂z + A(z) has a pole of order at most one at each of the removedpoints (this condition is generalized to a higher-dimensional variety X in astraightforward way by restricting the connection to all curves lying in X).

For example, consider the case when X = A× = Spec C[t, t−1] and assumethat the rank of E is equal to one, so it is a line bundle. We can embed X

into the projective curve P1, so that there are two points at infinity: t = 0and t =∞. Any holomorphic line bundle on A× can be trivialized. A generalconnection operator on A× then reads

∇∂t = ∂t +A(t), A(t) =M∑

i=N

Aiti. (1.2.5)

The group of invertible functions on A× consists of the elements αtn, α ∈C×, n ∈ Z. It acts on such operators by gauge transformations (1.2.4), andαtn acts as follows:

A(t) 7→ A(t)− n

t.

The set of equivalence classes of line bundles with a flat connection on A×

is in bijection with the quotient of the set of operators (1.2.5) by the groupof gauge transformations, or, equivalently, the quotient of C[t, t−1] by the

additive action of the group Z · 1t. Thus, we see that this set is huge.

Let us now look at connections with regular singularities only. The condi-tion that the connection (1.2.5) has a regular singularity at t = 0 means thatN ≥ −1. To see what the condition at t = ∞ is, we perform a change ofvariables u = t−1. Then, since ∂t = −u2∂u, we find that

∇∂u = ∂u − u−2A(u−1).

Hence the condition of regular singularity at ∞ is that M ≤ −1, and so tosatisfy both conditions, A(t) must have the form A(t) =

a

t.

Taking into account the gauge transformations, we conclude that the set ofequivalence classes of line bundles with a flat connection with regular singu-


larity on A× is isomorphic to C/Z. Given a connection (1.2.5) with A(t) =a

t,

the solutions of the equation (∂t +

a

t

)Φ(t) = 0

are ΦC(t) = C exp(−at), where C ∈ C. These are the horizontal sections ofour line bundle. Hence the monodromy of this connection along the loop goingcounterclockwise around the origin in A× is equal to exp(−2πia). We obtain aone-dimensional representation of π1(A×) ' Z sending 1 ∈ Z to exp(−2πia).Thus, the exponential map sets up a bijection between the set of equivalenceclasses of connections with regular singularities on A× and representations ofthe fundamental group of A×.

But if we allow irregular singularities, we obtain many more connections.

For instance, consider the connection ∂t+1t2

. It has an irregular singularity at

t = 0. The corresponding horizontal sections have the form C exp(−1/t), C ∈C, and hence the monodromy representation of π1(A×) corresponding to thisconnection is trivial, and so it is the same as for the connection ∇∂t

= ∂t.But this connection is not equivalent to the connection ∂t under the actionof the algebraic gauge transformations. We can of course obtain one fromthe other by the gauge action with the function exp(−1/t), but this functionis not algebraic. Thus, if we allow irregular singularities, there are manymore algebraic gauge equivalence classes of flat LG-connections than thereare homomorphisms from the fundamental group to LG.

1.2.6 Connections as the Langlands parameters

Our goal is to find a geometric enhancement of the set of equivalence classesof homomorphisms π1(D×) → LG, of which there are too few. The abovediscussion suggests a possible way to do it. We have seen that such homomor-phisms are the same as LG-bundles on D× with a connection with regularsingularity at the origin. Here by punctured disc we mean the schemeD× = Spec C((t)), so that the algebra of functions on D× is, by definition,C((t)). Any LG-bundle on D× can be trivialized, and so defining a connectionon it amounts to giving a first order operator

∂t +A(t), A(t) ∈ Lg((t)), (1.2.6)

where Lg is the Lie algebra of the Langlands dual group LG.Changing trivialization amounts to a gauge transformation (1.2.4) with

g ∈ LG((t)). The set of equivalence classes of LG-bundles with a connectionon D× is in bijection with the set of gauge equivalence classes of operators(1.2.6). We denote this set by LocLG(D×). Thus, we have

LocLG(D×) = ∂t +A(t), A(t) ∈ Lg((t))/LG((t)). (1.2.7)


A connection (1.2.6) has regular singularity if and only if A(t) has a pole oforder at most one at t = 0. If A−1 is the residue of A(t) at t = 0, then the cor-responding monodromy of the connection around the origin is an element ofLG equal to exp(2πiA−1). Two connections with regular singularity are (alge-braically) gauge equivalent to each other if and only if their monodromies areconjugate to each other. Hence the set of equivalence classes of connectionswith regular singularities is just the set of conjugacy classes of LG, or, equiv-alently, homomorphisms Z→ LG. These are the naive Langlands parametersthat we saw before (here we ignore the difference between the topologicaland algebraic fundamental groups). But now we generalize this by allowingconnections with arbitrary, that is, regular and irregular, singularities at theorigin. Then we obtain many more gauge equivalence classes.

We will often refer to points of LocLG(D×) as local systems on D×,meaning the de Rham version of the notion of a local system; namely, aprincipal bundle with a connection that has a pole of an arbitrary order atthe origin.

Thus, we come to the following proposal: the local Langlands parametersin the complex setting should be the points of LocLG(D×): the equivalenceclasses of flat LG-bundles on D× or, more concretely, the gauge equivalenceclasses (1.2.7) of first order differential operators.

We note that the Galois group of the local field Fq((t)) has a very intricatestructure: apart from the part coming from Gal(Fq/Fq) and the tame iner-tia, which is analogous to the monodromy that we observe in the complexcase, there is also the wild inertia subgroup whose representations are verycomplicated. It is an old idea that in the complex setting flat connectionswith irregular singularities in some sense play the role of representations ofthe Galois group whose restriction to the wild inertia subgroup is non-trivial.Our proposal simply exploits this idea in the context of the Langlands corre-spondence.

1.3 Representations of loop groups

Having settled the issue of the Langlands parameters, we have to decide whatit is that we will be parameterizing. Recall that in the classical setting thehomomorphism W ′F → LG parameterized irreducible smooth representationsof the group G(F ), F = Fq((t)). We start by translating this notion to therepresentation theory of loop groups.

1.3.1 Smooth representations

The loop group G((t)) contains the congruence subgroups

KN = g ∈ G[[t]] | g ≡ 1 mod tN, N ∈ Z+. (1.3.1)

1.3 Representations of loop groups 19

It is natural to call a representation of G((t)) on a complex vector space Vsmooth if for any vector v ∈ V there exists N ∈ Z+ such that KN · v = v.This condition may be interpreted as the continuity condition, if we define atopology on G((t)) by taking as the base of open neighborhoods of the identitythe subgroups KN , N ∈ Z+, as before.

But our group G is now a complex Lie group (not a finite group), andso G((t)) is an infinite-dimensional Lie group. More precisely, we view G((t))as an ind-group, i.e., as a group object in the category of ind-schemes. Atfirst glance, it is natural to consider the algebraic representations of G((t)).We observe that G((t)) is generated by the “parahoric” algebraic groups Pi,corresponding to the affine simple roots. For these subgroups the notion ofalgebraic representation makes perfect sense. A representation of G((t)) isthen said to be algebraic if its restriction to each of the Pi’s is algebraic.

However, this naive approach leads us to the following discouraging fact: anirreducible smooth representation of G((t)), which is algebraic, is necessarilyone-dimensional. To see that, we observe that an algebraic representation ofG((t)) gives rise to a representation of its Lie algebra g((t)) = g⊗C((t)), whereg is the Lie algebra of G. Representations of g((t)) obtained from algebraicrepresentations of G((t)) are called integrable. A smooth representation ofG((t)) gives rise to a smooth representation of g((t)), i.e., for any vector v thereexists N ∈ Z+ such that

g⊗ tNC[[t]] · v = 0. (1.3.2)

We can decompose g = gss⊕ r, where r is the center and gss is the semi-simplepart. An irreducible representation of g((t)) is therefore the tensor product ofan irreducible representation of gss and an irreducible representation of r((t)),which is necessarily one-dimensional since r((t)) is abelian. Now we have thefollowing

Lemma 1.3.1 A smooth integrable representation of the Lie algebra g((t)),where g is semi-simple, is trivial.

Proof. We follow the argument of [BD1], 3.7.11(ii). Let V be such arepresentation. Then there exists N ∈ Z+ such that (1.3.2) holds. Let uspick a Cartan decomposition g = n− ⊕ h ⊕ n+ (see Appendix A.3 below)of the constant subalgebra g ⊂ g((t)). Let H be the Cartan subgroup of Gcorresponding to h ⊂ g. By our assumption, V is an algebraic representationof H, and hence it decomposes into a direct sum of weight spaces V =

⊕Vχ

over the integral weights χ of H. Thus, any vector v ∈ V may be decomposedas the sum v =

∑χ∈h∗ vχ, where vχ ∈ Vχ.

Now observe that there exists an element h ∈ H((t)) such that

n+ ⊂ h(g⊗ tNC[[t]])h−1.


Then we find that the vector h · v is invariant under n+. Since the action ofn+ is compatible with the grading by the integral weights of H, we obtainthat each vector h · vχ is also invariant under n+. Consider the g-submoduleof V generated by the vector h · vχ. By our assumption, the action of g on Vintegrates to an algebraic representation of G on V . Hence V is a direct sumof finite-dimensional irreducible representations of G with dominant integralweights. Therefore h · vχ = 0, and hence vχ = 0, unless χ is a dominantintegral weight. Next, observe that h−1(g⊗ tNC[[t]])h contains n−. Applyingthe same argument to h−1 ·v, we find that vχ = 0 unless χ is an anti-dominantintegral weight.

Therefore we obtain that v = v0, and hence h · v is invariant under h⊕ n+.But then the g-submodule generated by h·v is trivial. Hence h·v is g-invariant,and so v is h−1gh-invariant. But the Lie algebras g⊗tNC[[t]] and h−1gh, whereh runs over those elements of H((t)) for which h(g⊗ tNC[[t]])h−1 contains n+,generate the entire Lie algebra g((t)). Therefore v is g((t))-invariant. Weconclude that V is a trivial representation of g((t)).

Thus, we find that the class of algebraic representations of loop groups turnsout to be too restrictive. We could relax this condition and consider differ-entiable representations, i.e., the representations of G((t)) considered as a Liegroup. But it is easy to see that the result would be the same. Replacing G((t))by its central extension G would not help us much either: irreducible inte-grable representations of G are parameterized by dominant integral weights,and there are no extensions between them [K2]. These representations areagain too sparse to be parameterized by the geometric data considered above.Therefore we should look for other types of representations.

Going back to the original setup of the local Langlands correspondence,we recall that there we considered representations of G(Fq((t))) on C-vectorspaces, so we could not possibly use the algebraic structure of G(Fq((t))) as anind-group over Fq. Therefore we cannot expect the class of algebraic (or dif-ferentiable) representations of the complex loop group G((t)) to be meaningfulfrom the point of view of the Langlands correspondence. We should view theloop group G((t)) as an abstract topological group, with the topology definedby means of the congruence subgroups; in other words, consider its smoothrepresentations as an abstract group.

To give an example of such a representation, consider the vector spaceof finite linear combinations

∑δx, where x runs over the set of points of the

quotient G((t))/KN . The group G((t)) naturally acts on this space: g (∑δx) =∑

δg·x. It is clear that for any g ∈ G((t)) the subgroup gKNg−1 contains KM

for large enough M , and so this is indeed a smooth representation of G((t)).But it is certainly not an algebraic representation (nor is it differentiable).In fact, it has absolutely nothing to do with the algebraic structure of G((t)),which is a serious drawback. After all, the whole point of trying to generalize


the Langlands correspondence from the setting of finite fields to that of thecomplex field was to be able to use the powerful tools of complex algebraicgeometry. If our representations are completely unrelated to geometry, thereis not much that we can learn from them.

So we need to search for some geometric objects that encapsulate repre-sentations of our groups and make sense both over a finite field and over thecomplex field.

1.3.2 From functions to sheaves

We start by revisiting smooth representations of the group G(F ), where F =Fq((t)). We realize such representations more concretely by considering theirmatrix coefficients. Let (V, π) be an irreducible smooth representation ofG(F ). We define the contragredient representation V ∨ as the linear span ofall smooth vectors in the dual representation V ∗. This span is stable under theaction of G(F ) and so it admits a smooth representation (V ∨, π∨) of G(F ).Now let φ be a KN -invariant vector in V ∨. Then we define a linear map

V → C(G(F )/KN ), v 7→ fv,

where

fv(g) = 〈π∨(g)φ, v〉.

Here C(G(F )/KN ) denotes the vector space of C-valued locally constant func-tions on G(F )/KN . The group G(F ) naturally acts on this space by theformula (g · f)(h) = f(g−1h), and the above map is a morphism of represen-tations, which is non-zero, and hence injective, if (V, π) is irreducible.

Thus, we realize our representation in the space of functions on the quo-tient G(F )/KN . More generally, we may realize representations in spaces offunctions on the quotient G((t))/K with values in a finite-dimensional vectorspace, by considering a finite-dimensional subrepresentation of K inside Vrather than the trivial one.

An important observation here is that G(F )/K, where F = Fq((t)) and K

is a compact subgroup of G(F ), is not only a set, but it is the set of points ofan algebraic variety (more precisely, an ind-scheme) defined over the field Fq.For example, for K0 = G(Fq[[t]]), which is the maximal compact subgroup,the quotient G(F )/K0 is the set of Fq-points of the ind-scheme called theaffine Grassmannian.

Next, we recall an important idea going back to Grothendieck that functionson the set of Fq-points on an algebraic variety X defined over Fq can often beviewed as the “shadows” of the so-called `-adic sheaves on X.

Let us discuss them briefly. Let ` be a prime that does not divide q. Thedefinition of the category of `-adic sheaves on X involves several steps (see,e.g., [Mi, FK]). First we consider locally constant Z/`mZ-sheaves on X in the


etale topology (in which the role of open subsets is played by etale morphismsU → X). A Z`-sheaf on X is by definition a system (Fm) of locally constantZ/`mZ-sheaves satisfying natural compatibilities. Then we define the categoryof Q`-sheaves by killing the torsion sheaves in the category of Z`-sheaves. Ina similar fashion we define the category of E-sheaves on X, where E is afinite extension of Q`. Finally, we take the direct limit of the categoriesof E-sheaves on X, and the objects of this category are called the locallyconstant `-adic sheaves on X. Such a sheaf of rank n is the same as an n-dimensional representation of the Galois group of the field of functions on Xthat is everywhere unramified. Thus, locally constant `-adic sheaves are theanalogues of the usual local systems (with respect to the analytic topology)in the case of complex algebraic varieties.

We may generalize the above definition of an `-adic local system on X byallowing the Z/`nZ-sheaves Fn to be constructible, i.e., for which there existsa stratification of X by locally closed subvarieties Xi such that the sheavesF|Xi

are locally constant. As a result, we obtain the notion of a constructible`-adic sheaf on X, or an `-adic sheaf, for brevity.

The key step in the geometric reformulation of this notion is the Grothen-dieck fonctions-faisceaux dictionary (see, e.g., [La]). Let F be an `-adicsheaf and x be an Fq1-point of X, where q1 = qm. Then we have the Frobeniusconjugacy class Frx acting on the stalk Fx of F at x. Hence we can define afunction fq1(F) on the set of Fq1-points of X, whose value at x is Tr(Frx,Fx).This function takes values in the algebraic closure Q` of Q`. But there isnot much of a difference between Q`-valued functions and C-valued functions:since they have the same cardinality, Q` and C may be identified as abstractfields. Besides, in most interesting cases, the values actually belong to Q,which is inside both Q` and C.

More generally, if F is a complex of `-adic sheaves, we define a functionfq1(F) on X(Fq1) by taking the alternating sums of the traces of Frx on thestalk cohomologies of F at x. The map F → fq1(F) intertwines the naturaloperations on sheaves with natural operations on functions (see [La], Section1.2). For example, pull-back of a sheaf corresponds to the pull-back of afunction, and push-forward of a sheaf with compact support corresponds tothe fiberwise integration of a function (this follows from the Grothendieck–Lefschetz trace formula).

Let K0(ShX) be the complexified Grothendieck group of the category of`-adic sheaves on X. Then the above construction gives us a map

K0(ShX)→∏m≥1

X(Fqm),

and it is known that this map is injective (see [La]).Therefore we may hope that the functions on the quotients G(F )/KN ,


which realize our representations, arise, via this construction, from `-adicsheaves, or, more generally, from complexes of `-adic sheaves, on X.

Now, the notion of a constructible sheaf (unlike the notion of a function)has a transparent and meaningful analogue for a complex algebraic variety X;namely, those sheaves of C-vector spaces whose restrictions to the strata of astratification of the variety X are locally constant. The affine Grassmannianand more general ind-schemes underlying the quotients G(F )/KN may bedefined both over Fq and C. Thus, it is natural to consider the categories ofsuch sheaves (or, more precisely, their derived categories) on these ind-schemesover C as the replacements for the vector spaces of functions on their pointsrealizing smooth representations of the group G(F ).

We therefore naturally come to the idea, advanced in [FG2], that the rep-resentations of the loop group G((t)) that we need to consider are not realizedon vector spaces, but on categories, such as the derived category of coher-ent sheaves on the affine Grassmannian. Of course, such a category has aGrothendieck group, and the group G((t)) will act on the Grothendieck groupas well, giving us a representation of G((t)) on a vector space. But we obtainmuch more structure by looking at the categorical representation. The objectsof the category, as well as the action, will have a geometric meaning, and thuswe will be using the geometry as much as possible.

Let us summarize: to each local Langlands parameter χ ∈ LocLG(D×) wewish to attach a category Cχ equipped with an action of the loop group G((t)).But what kind of categories should these Cχ be and what properties do weexpect them to satisfy?

To get closer to answering these questions, we wish to discuss two more stepsthat we can make in the above discussion to get to the types of categories withan action of the loop group that we will consider in this book.

1.3.3 A toy model

At this point it is instructive to detour slightly and consider a toy modelof our construction. Let G be a split reductive group over Z, and B itsBorel subgroup. A natural representation of G(Fq) is realized in the space ofcomplex- (or Q`-) valued functions on the quotient G(Fq)/B(Fq). It is naturalto ask what is the “correct” analogue of this representation if we replace thefield Fq by the complex field and the group G(Fq) by G(C). This may beviewed as a simplified version of our quandary, since instead of consideringG(Fq((t))) we now look at G(Fq).

The quotient G(Fq)/B(Fq) is the set of Fq-points of the algebraic varietydefined over Z called the flag variety of G and denoted by Fl. Our discussionin the previous section suggests that we first need to replace the notion of


a function on Fl(Fq) by the notion of an `-adic sheaf on the variety FlFq =Fl⊗

ZFq.

Next, we replace the notion of an `-adic sheaf on Fl, considered as analgebraic variety over Fq, by the notion of a constructible sheaf on FlC =Fl⊗

ZC, which is an algebraic variety over C. The complex algebraic group GC

naturally acts on FlC and hence on this category. Now we make two morereformulations of this category.

First of all, for a smooth complex algebraic variety X we have a Riemann–Hilbert correspondence, which is an equivalence between the derived cat-egory of constructible sheaves on X and the derived category of D-moduleson X that are holonomic and have regular singularities.

Here we consider the sheaf of algebraic differential operators on X andsheaves of modules over it, which we simply refer to as D-modules. The sim-plest example of a D-module is the sheaf of sections of a vector bundle onX equipped with a flat connection. The flat connection enables us to mul-tiply any section by a function and we can use the flat connection to act onsections by vector fields. The two actions generate an action of the sheaf ofdifferential operators on the sections of our bundle. The sheaf of horizontalsections of this bundle is then a locally constant sheaf on X. We have seenabove that there is a bijection between the set of isomorphism classes of rankn bundles on X with connection having regular singularities and the set ofisomorphism classes of locally constant sheaves on X of rank n, or, equiva-lently, n-dimensional representations of π1(X). This bijection may be elevatedto an equivalence of the corresponding categories, and the general Riemann–Hilbert correspondence is a generalization of this equivalence of categoriesthat encompasses more general D-modules.

The Riemann–Hilbert correspondence allows us to associate to any holo-nomic D-module on X a complex of constructible sheaves on X, and this givesus a functor between the corresponding derived categories, which turns outto be an equivalence if we restrict ourselves to the holonomic D-modules withregular singularities (see [Bor2, GM] for more details).

Thus, over C we may pass from constructible sheaves to D-modules. In ourcase, we consider the category of (regular holonomic) D-modules on the flagvariety FlC. This category carries a natural action of GC.

Finally, let us observe that the Lie algebra g of GC acts on the flag varietyinfinitesimally by vector fields. Therefore, given a D-module F on FlC, thespace of its global sections Γ(FlC,F) has the structure of g-module. Weobtain a functor Γ from the category of D-modules on FlC to the categoryof g-modules. A. Beilinson and J. Bernstein have proved that this functor isan equivalence between the category of all D-modules on FlC (not necessarilyregular holonomic) and the category C0 of g-modules on which the center of theuniversal enveloping algebra U(g) acts through the augmentation character.


Thus, we can now answer our question as to what is a meaningful geomet-ric analogue of the representation of the finite group G(Fq) on the space offunctions on the quotient G(Fq)/B(Fq). The answer is the following: it is acategory equipped with an action of the algebraic group GC. This categoryhas two incarnations: one is the category of D-modules on the flag varietyFlC, and the other is the category C0 of modules over the Lie algebra g withtrivial central character. Both categories are equipped with natural actionsof the group GC.

Let us pause for a moment and spell out what exactly we mean when we saythat the group GC acts on the category C0. For simplicity, we will describe theaction of the corresponding group G(C) of C-points of GC.† This means thefollowing: each element g ∈ G gives rise to a functor Fg on C0 such that F1 isthe identity functor, and the functor Fg−1 is quasi-inverse to Fg. Moreover, forany pair g, h ∈ G we have a fixed isomorphism of functors ig,h : Fgh → Fg Fh

so that for any triple g, h, k ∈ G we have the equality ih,kig,hk = ig,high,k ofisomorphisms Fghk → Fg Fh Fk. (We remark that the last condition couldbe relaxed: we could ask only that ih,kig,hk = γg,h,kig,high,k, where γg,h,k is anon-zero complex number for each triple g, h, k ∈ G; these numbers then mustsatisfy a three-cocycle condition. However, we will only consider the situationwhere γg,h,k ≡ 1.)

The functors Fg are defined as follows. Given a representation (V, π) of g

and an element g ∈ G(C), we define a new representation Fg((V, π)) = (V, πg),where by definition πg(x) = π(Adg(x)). Suppose that (V, π) is irreducible.Then it is easy to see that (V, πg) ' (V, π) if and only if (V, π) is integrable,i.e., is obtained from an algebraic representation of G. This is equivalentto this representation being finite-dimensional. But a general representation(V, π) is infinite-dimensional, and so it will not be isomorphic to (V, πg), atleast for some g ∈ G.

Now we consider morphisms in C0, which are just g-homomorphisms. Givena g-homomorphism between representations (V, π) and (V ′, π′), i.e., a linearmap T : V → V ′ such that Tπ(x) = π′(x)T for all x ∈ g, we set Fg(T ) = T .The isomorphisms ig,h are all equal to the identity in this case.

The simplest examples of objects of the category C0 are the Verma modulesinduced from one-dimensional representations of a Borel subalgebra b ⊂ g.The corresponding D-module is the D-module of “delta-functions” supportedat the point of FlC stabilized by b. In the Grothendieck group of the categoryC0 the classes of these objects span a subrepresentation of G(C) (considerednow as a discrete group!), which looks exactly like the representation we de-fined at the beginning of Section 1.3.2. What we have achieved is that we have

† More generally, for any C-algebra R, we have an action of G(R) on the correspondingbase-changed category over R. Thus, we are naturally led to the notion of an algebraicgroup (or, more generally, a group scheme) acting on an abelian category, which is spelledout in [FG2], Section 20.


replaced this representation by something that makes sense from the point ofview of the representation theory of the complex algebraic group G (ratherthan the corresponding discrete group); namely, the category C0.

1.3.4 Back to loop groups

In our quest for a complex analogue of the local Langlands correspondence weneed to decide what will replace the notion of a smooth representation of thegroup G(F ), where F = Fq((t)). As the previous discussion demonstrates, weshould consider representations of the complex loop group G((t)) on variouscategories of D-modules on the ind-schemes G((t))/K, where K is a “compact”subgroup of G((t)), such as G[[t]] or the Iwahori subgroup (the preimage of aBorel subgroup B ⊂ G under the homomorphism G[[t]] → G), or the cate-gories of representations of the Lie algebra g((t)). Both scenarios are viable,and they lead to interesting results and conjectures, which we will discuss indetail in Chapter 10, following [FG2]. In this book we will concentrate on thesecond scenario and consider categories of (projective) modules ove the loopalgebra g((t)).

The group G((t)) acts on the category of representations of g((t)) in the waythat we described in the previous section. In order to make the correspondingrepresentation of G((t)) smooth, we need to restrict ourselves to those repre-sentations on which the action of the Lie subalgebra g⊗ tNC[[t]] is integrablefor some N > 0. Indeed, on such representations the action of g ⊗ tNC[[t]]may be exponentiated to an action of its Lie group, which is the congruencesubgroup KN . If (V, π) is such a representation, then for each g ∈ KN theoperator of the action of g on V will provide an isomorphism between (V, π)and Fg((V, π)). Therefore we may say that (V, π) is “stable” under Fg.

As the following lemma shows, this condition is essentially equivalent tothe condition of a g((t))-module being smooth, i.e., such that any vector isannihilated by the Lie algebra g⊗ tMC[[t]] for sufficiently large M .

Lemma 1.3.1 Suppose that (V, π) is a smooth finitely generated module overthe Lie algebra g((t)). Then there exists N > 0 such that the action of the Liesubalgebra g⊗ tNC[[t]] on V is integrable.

Proof. Let v1, . . . , vk be a generating set of vectors in V . Since (V, π)is smooth, each vector vi is annihilated by a Lie subalgebra g ⊗ tNiC[[t]] forsome Ni. Let Mi be the induced representation

Mi = Indg((t))

g⊗tNi C[[t]]C = U(g((t))) ⊗

g⊗tNi C[[t]]C.

Then we have a surjective homomorphism⊕k

i=1Mi → V sending the gener-ating vector of Mi to vi for each i = 1, . . . , k. Let N be the largest number


among N1, . . . , Nk. Then the action of g⊗ tNC[[t]] on each Mi is integrable,and hence the same is true for V .

1.3.5 From the loop algebra to its central extension

Thus, we can take as the categorical analogue of a smooth representation ofa reductive GOP over a local non-archimedian field the category of smoothfinitely generated modules over the Lie algebra g((t)) equipped with a naturalaction of the loop group G((t)). (In what follows we will drop the condition ofbeing finitely generated.)

Let us observe however that we could choose instead the category of smoothrepresentations of a central extension of g((t)). The group G((t)) still acts onsuch a central extension via the adjoint action. Since the action of the groupG((t)) on the category comes through its adjoint action, no harm will be doneif we extend g((t)) by a central subalgebra.

The notion of a central extension of a Lie algebra is described in detailin Appendix A.4. In particular, it is explained there that the equivalenceclasses of central extensions of g((t)) are described by the second cohomologyH2(g((t)),C).†

We will use the decomposition g = gss ⊕ r, where r is the center and gss isthe semi-simple part. Then it is possible to show that

H2(g((t)),C) ' H2(gss((t)),C)⊕H2(r((t)),C).

In other words, the central extension of g((t)) is determined by its restrictionto gss((t)) and to r((t)). The central extensions that we will consider will betrivial on the abelian part r((t)), and so our central extension will be a directsum gss⊕r((t)). An irreducible representation of this Lie algebra is isomorphicto the tensor product of an irreducible representation of gss and that of r((t)).But the Lie algebra r((t)) is abelian, and so its irreducible representations areone-dimensional. It is not hard to deal with these representations separately,and so from now on we will focus on representations of central extensions ofg((t)), where g is a semi-simple Lie algebra.

If g is semi-simple, it can be decomposed into a direct sum of simple Liealgebras gi, i = 1, . . . ,m, and we have

H2(g((t)),C) 'm⊕

i=1

H2(gi((t)),C).

Again, without loss of generality we can treat each simple factor gi separately.

† Here we need to take into account our topology on the loop algebra g((t)) in which thebase of open neighborhoods of zero is given by the Lie subalgebras g⊗ tN C[[t]], N ∈ Z+.Therefore we should restrict ourselves to the corresponding continuous cohomology. Wewill use the same notation H2(g((t)), C) for it.


Hence from now on we will assume that g is a simple Lie algebra. In this casewe have the following description of H2(g((t)),C).

Recall that a inner product κ on a Lie algebra g is called invariant if

κ([x, y], z) + κ(y, [x, z]) = 0, ∀x, y, z ∈ g.

This formula is the infinitesimal version of the formula κ(Ad g · y,Ad g · z) =κ(y, z) expressing the invariance of the form with respect to the Lie groupaction (with g being an element of the corresponding Lie group). The vectorspace of non-degenerate invariant inner products on a finite-dimensional sim-ple Lie algebra g is one-dimensional. One can produce such a form startingwith any non-trivial finite-dimensional representation ρV : g→ EndV of g bythe formula

κV (x, y) = TrV (ρV (x)ρV (y)).

The standard choice is the adjoint representation, which gives rise to theKilling form κg. The following result is well-known.

Lemma 1.3.2 For a simple Lie algebra g the space H2(g((t)),C) is one-dimensional and is identified with the space of invariant bilinear forms on g.Given such a form κ, the corresponding central extension can be constructedfrom the cocycle

c(A⊗ f(t), B ⊗ g(t)) = −κ(A,B) Rest=0 fdg. (1.3.3)

Here for a formal Laurent power series a(t) =∑

n∈Z antn we set

Rest=0 a(t)dt = a−1.

1.3.6 Affine Kac–Moody algebras and their representations

The central extension

0→ C1→ gκ → g((t))→ 0

corresponding to a non-zero cocycle κ is called the affine Kac–Moody al-gebra . We denote it by gκ. It is customary to refer to κ as the level. As avector space, it is equal to the direct sum g((t)) ⊕ C1, and the commutationrelations read

[A⊗ f(t), B ⊗ g(t)] = [A,B]⊗ f(t)g(t)− (κ(A,B) Res fdg)1, (1.3.4)

where 1 is a central element, which commutes with everything else. Notethat the Lie algebra gκ and gκ′ are isomorphic for non-zero inner productsκ, κ′. Indeed, in this case we have κ = λκ′ for some λ ∈ C×, and the mapgκ → gκ′ , which is equal to the identity on g((t)) and sends 1 to λ1, is an


isomorphism. By Lemma 1.3.2, the Lie algebra gκ with non-zero κ is in facta universal central extension of g((t)). (The Lie algebra g0 is by definition thesplit extension g((t))⊕ C1.)

Note that the restriction of the cocycle (1.3.3) to the Lie subalgebra g ⊗tNC[[t]], N ∈ Z+ is equal to 0, and so it remains a Lie subalgebra of gκ.A smooth representation of gκ is a representation such that every vectoris annihilated by this Lie subalgebra for sufficiently large N . Note that thestatement of Lemma 1.3.1 remains valid if we replace g((t)) by gκ.

Thus, we define the category gκ -mod whose objects are smooth gκ-moduleson which the central element 1 acts as the identity. The morphisms are homo-morphisms of representations of gκ. Throughout this book, unless specifiedotherwise, by a “gκ-module” we will always mean a module on which thecentral element 1 acts as the identity.†

The group G((t)) acts on the Lie algebra gκ for any κ. Indeed, the adjointaction of the central extension of G((t)) factors through the action of G((t)).It is easy to compute this action and to find that

g · (A(t) + c1) =(gA(t)g−1 + Rest=0 κ((∂tg)g−1, A(t))1

).

It is interesting to observe that the dual space to gκ (more precisely, a hyper-plane in the dual space) may be identified with the space of connections onthe trivial G-bundle on D× so that the coadjoint action gets identified withthe gauge action of G((t)) on the space of such connections (for more on this,see [FB], Section 16.4).

We use the action of G((t)) on gκ to construct an action of G((t)) on thecategory gκ -mod, in the same way as in Section 1.3.3. Namely, suppose weare given an object (M,ρ) of gκ -mod, where M is a vector space and ρ : gκ →EndM is a Lie algebra homomorphism making M into a smooth gκ-module.Then, for each g ∈ G((t)), we define a new object (M,ρg) of gκ -mod, whereby definition ρg(x) = π(Adg(x)). If we have a morphism (M,ρ) → (M ′, ρ′)in gκ -mod, then we obtain an obvious morphism (M,ρg) → (M ′, ρ′g). Thus,we have defined a functor Fg : gκ -mod → gκ -mod. These functors satisfythe conditions of Section 1.3.3. Thus, we obtain an action of G((t)) on thecategory gκ -mod.

Recall the space LocLG(D×) of the Langlands parameters that we definedin Section 1.2.6. Elements of LocLG(D×) have a concrete description as gaugeequivalence classes of first order operators ∂t + A(t), A(t) ∈ Lg((t)), modulothe action of LG((t)) (see formula (1.2.7)).

We can now formulate the local Langlands correspondence over C as thefollowing problem:

† Note that we could have 1 act instead as λ times the identity for λ ∈ C×; but thecorresponding category would just be equivalent to the category bgλκ -mod.


To each local Langlands parameter χ ∈ LocLG(D×) associatea subcategory gκ -modχ of gκ -mod which is stable under theaction of the loop group G((t)).

We wish to think of the category gκ -mod as “fibering” over the space oflocal Langlands parameters LocLG(D×), with the categories gκ -modχ beingthe “fibers” and the group G((t)) acting along these fibers. From this pointof view the categories gκ -modχ should give us a “spectral decomposition” ofthe category gκ -mod over LocLG(D×).

In Chapter 10 of this book we will present a concrete proposal made in [FG2]describing these categories in the special case when κ = κc is the critical level,which is minus one half of κg, the Killing form defined above. This proposal isbased on the fact that at the critical level the center of the category gκc

-mod,which is the same as the center of the completed universal enveloping algebraof gκc , is very large. It turns out that its spectrum is closely related to thespace LocLG(D×) of Langlands parameters, and this will enable us to definegκc -modχ as, roughly speaking, the category of smooth gκc -modules with afixed central character.

In order to explain more precisely how this works, we need to develop therepresentation theory of affine Kac–Moody algebras and in particular describethe structure of the center of the completed enveloping algebra of gκc

. In thenext chapter we will start on a long journey towards this goal. This will occupythe main part of this book. Then in Chapter 10 we will show, following thepapers [FG1]–[FG6], how to use these results in order to construct the localgeometric Langlands correspondence for loop groups.

2

Vertex algebras

Let g be a simple finite-dimensional Lie algebra and gκ the correspondingaffine Kac–Moody algebra (the central extension of g((t))), introduced in Sec-tion 1.3.6. We have the category gκ -mod, whose objects are smooth gκ-modules on which the central element 1 acts as the identity. As explained atthe end of the previous chapter, we wish to show that this category “fibers”over the space of Langlands parameters, which are gauge equivalence classesof LG-connections on the punctured disc D× (or perhaps, something similar).Moreover, the loop group G((t)) should act on this category “along the fibers.”

Any abelian category may be thought of as “fibering” over the spectrum ofits center. Hence the first idea that comes to mind is to describe the center ofthe category gκ -mod in the hope that its spectrum is related to the Langlandsparameters. As we will see, this is indeed the case for a particular value of κ.

In order to show that, however, we first need to develop a technique fordealing with the completed universal enveloping algebra of gκ. This is theformalism of vertex algebras. In this chapter we will first motivate the neces-sity of vertex algebras and then introduce the basics of the theory of vertexalgebras.

2.1 The center

2.1.1 The case of simple Lie algebras

Let us first recall what is the center of an abelian category. Let C be an abeliancategory over C. The center Z(C) is by definition the set of endomorphismsof the identity functor on C. Let us recall that such an endomorphism is asystem of endomorphisms eM ∈ HomC(M,M), for each object M of C, whichis compatible with the morphisms in C: for any morphism f : M → N in Cwe have f eM = eN f . It is clear that Z(C) has a natural structure of acommutative algebra over C.

Let S = SpecZ(C). This is an affine algebraic variety such that Z(C) is thealgebra of functions on S. Each point s ∈ S defines an algebra homomorphism

31

32 Vertex algebras

(equivalently, a character) ρs : Z(C)→ C (evaluation of a function at the points). We define the full subcategory Cs of C whose objects are the objects of Con which Z(C) acts according to the character ρs. It is instructive to think ofthe category C as “fibering” over S, with the fibers being the categories Cs.

Now suppose that C = A -mod is the category of left modules over anassociative C-algebra A. Then A itself, considered as a left A-module, is anobject of C, and so we obtain a homomorphism

Z(C)→ Z(EndAA) = Z(Aopp) = Z(A),

where Z(A) is the center of A. On the other hand, each element of Z(A)defines an endomorphism of each object of A -mod, and so we obtain a homo-morphism Z(A) → Z(C). It is easy to see that these maps define mutuallyinverse isomorphisms between Z(C) and Z(A).

If g is a Lie algebra, then the category g -mod of g-modules coincides withthe category U(g) -mod of U(g)-modules, where U(g) is the universal envelop-ing algebra of g (see Appendix A.2). Therefore the center of the categoryg -mod is equal to the center of U(g), which by abuse of notation we denoteby Z(g).

Let us recall the description of Z(g) in the case when g is a finite-dimensionalsimple Lie algebra over C of rank ` (see Section A.3). It can be proved by acombination of results of Harish-Chandra and Chevalley (see [Di]).

Theorem 2.1.1 The center Z(g) is a polynomial algebra C[Pi]i=1,...,` gener-ated by elements Pi, i = 1, . . . , `, of orders di + 1, where di are the exponentsof g.

An element P of U(g) is said to have order i if it belongs to the ith termU(g)≤i of the Poincare–Birkhoff–Witt filtration on U(g) described in Ap-pendix A.2, but does not belong to U(g)≤(i−1).

The exponents form a set of positive integers attached to each simple Liealgebra. For example, for g = sln this set is 1, . . . , n− 1. There are severalequivalent definitions, and the above theorem may be taken as one of them(we will encounter another definition in Section 4.2.4 below).

The first exponent of any simple Lie algebra g is always 1, so there is alwaysa quadratic element in the center of U(g). This element is called the Casimirelement and can be constructed as follows. Let Ja be a basis for g as avector space. Fix any non-zero invariant inner product κ0 on g and let Jabe the dual basis to Ja with respect to κ0. Then the Casimir element isgiven by the formula

P =12

dim g∑a=1

JaJa.

2.1 The center 33

Note that it does not depend on the choice of the basis Ja. Moreover,changing κ0 to κ′0 would simply multiply P by a scalar λ such that κ = λκ′0.

It is a good exercise to compute this element in the case of g = sl2. Wehave the standard generators of sl2

e =(

0 10 0

), h =

(1 00 −1

), f =

(0 01 0

)and the inner product

κ0(a, b) = Tr ab.

Then the Casimir element is

P =12

(ef + fe+

12h2

).

2.1.2 The case of affine Lie algebras

We wish to obtain a result similar to Theorem 2.1.1 describing the universalenveloping algebra of an affine Kac–Moody algebra.

The first step is to define an appropriate enveloping algebra whose cat-egory of modules coincides with the category gκ -mod. Let us recall fromSection 1.3.6 that objects of gκ -mod are gκ-modules M on which the centralelement 1 acts as the identity and which are smooth, that is, for any vectorv ∈M we have

(g⊗ tNC[[t]]) · v = 0 (2.1.1)

for sufficiently large N .As a brief aside, let us remark that we have a polynomial version gpol

κ of theaffine Lie algebra, which is the central extension of the polynomial loop algebrag[t, t−1]. Let gpol

κ -mod be the category of smooth gpolκ -modules, defined in the

same way as above, with the Lie algebra g⊗tNC[t] replacing g⊗tNC[[t]]. Thesmoothness condition allows us to extend the action of gpol

κ on any object ofthis category to an action of gκ. Therefore the categories gpol

κ and gκ -modcoincide.

Here it is useful to explain why we prefer to work with the Lie algebra gκ asopposed to gpol

κ . This is because gκ is naturally attached to the (formal) punc-tured discD× = Spec C((t)), whereas as gpol

κ is attached to C× = Spec C[t, t−1](or the unit circle). This means in particular that, unlike gpol

κ , the Lie alge-bra gκ is not tied to a particular coordinate t, but we may replace t by anyother formal coordinate on the punctured disc. This means, as we will doin Section 3.5.2, that gκ may be attached to the formal neighborhood of anypoint on a smooth algebraic curve, a property that is very important in ap-plications that we have in mind, such as passing from the local to global

34 Vertex algebras

Langlands correspondence. In contrast, the polynomial version gpolκ is forever

tied to C×.Going back to the category gκ -mod, we see that there are two properties

that its objects satisfy. Therefore it does not coincide with the category of allmodules over the universal enveloping algebra U(gκ) (which is the categoryof all gκ-modules). We need to modify this algebra.

First of all, since 1 acts as the identity, the action of U(gκ) factors throughthe quotient

Uκ(g) def= U(gκ)/(1− 1).

Second, the smoothness condition (2.1.1) implies that the action of Uκ(g)extends to an action of its completion defined as follows.

Define a linear topology on Uκ(g) by using as the basis of neighborhoodsfor 0 the following left ideals:

IN = Uκ(g)(g⊗ tNC[[t]]), N > 0.

Let Uκ(g) be the completion of Uκ(g) with respect to this topology. Wecall it that completed universal enveloping algebra of gκ. Note that,equivalently, we can write

Uκ(g) = lim←−

Uκ(g)/IN .

Even though the IN ’s are only left ideals (and not two-sided ideals), one checksthat the associative product structure on Uκ(g) extends by continuity to anassociative product structure on Uκ(g) (this follows from the fact that theLie bracket on Uκ(g) is continuous in the above topology). Thus, Uκ(g) is acomplete topological algebra. It follows from the definition that the categorygκ -mod coincides with the category of discrete modules over Uκ(g) on whichthe action of Uκ(g) is pointwise continuous (this is precisely equivalent to thecondition (2.1.1)).

It is now easy to see that the center of our category gκ -mod is equal to thecenter of the algebra Uκ(g), which we will denote by Zκ(g). The argumentis similar to the one we used above: though Uκ(g) itself is not an objectof gκ -mod, we have a collection of objects Uκ(g)/IN . Using this collection,we obtain an isomorphism between the center of the category gκ -mod andthe inverse limit of the algebras Z(Endbgκ

Uκ(g)/IN ), which, by definition,coincides with Zκ(g).

Now we can formulate our first question:

describe the center Zκ(g) for all levels κ.

We will see that the center Zκ(g) is trivial (i.e., equal to the scalars) unlessκ = κc, the critical level. However, at the critical level Zκc

(g) is large, and its

2.1 The center 35

structure is reminiscent to that of Z(g) described in Theorem 2.1.1. For theprecise statement, see Theorem 4.3.6.

2.1.3 The affine Casimir element

Before attempting to describe the entire center, let us try to construct somecentral elements “by hand.” In the finite-dimensional case the simplest gen-erator of Z(g) was particularly easy to construct. So we start by attemptingto define a similar operator in the affine case.

Let A be any element of g and n an integer. Then A ⊗ tn is an elementof g ⊗ C((t)) and hence of gκ. We denote this element by An. We collect allof the elements associated to A ∈ g into a single formal power series in anauxiliary variable z:

A(z) =∑n∈Z

Anz−n−1.

The shift by one in the exponent for z may seem a little strange at first butit is convenient to have. For example, we have the following formula

An = Resz=0A(z)zndz.

Note that this is just a formal notation. None of the power series we useactually has to converge anywhere.

Now, an obvious guess for an equivalent to the Casimir operator is theformal power series

12

dim g∑a=1

Ja(z)Ja(z). (2.1.2)

There are, however, many problems with this expression. If we extract thecoefficients of this sum, we see that they are two-way infinite sums. Theinfinity, by itself, is not a problem, because our completed enveloping algebraUκ(g) contains infinite sums. But, unfortunately, here we encounter a “wronginfinity,” which needs to be corrected.

To be more concrete, let us look in detail at the case of g = sl2. Then ourpotential Casimir element is

P (z) =12

(e(z)f(z) + f(z)e(z) +

12h(z)h(z)

). (2.1.3)

It is easy to write down the coefficients in front of particular powers of z inthis series. If we write

P (z) =∑n∈Z

PNz−N−2,

then

PN =∑

m+n=N

(emfn + fmen +12hmhn). (2.1.4)

36 Vertex algebras

None of these expressions belongs to Uκ(sl2).To see this, let us observe that, by definition, an element of Uκ(sl2) may be

written in the form

K +∑n≥0

(Qnen +Rnfn + Snhn),

where K,Qn, Rn, Sn are finite linear combinations of monomials in the gen-erators em, fm, hm,m ∈ Z.

Let us examine the first term in (2.1.4). It may be written as the sum oftwo terms ∑

n+m=N ;n≥0

emfn +∑

n+m=N ;n<0

emfn. (2.1.5)

The first of them belongs to Uκ(sl2), but the second one does not: the orderof the two factors is wrong! This means that this element does not give riseto a well-defined operator on a module from the category sl2,κ -mod. Indeed,we can write

emfn = fnem + [em, fn] = fnem + hm+n.

Thus, the price to pay for switching the order is the commutator between thetwo factors, which is non-zero. Therefore, while the sum∑

n+m=N ;n<0

fnem

belongs to Uκ(sl2) and its action is well-defined on any module from sl2,κ -mod,the sum ∑

n+m=N ;n<0

emfn

that we are given differs from it by hm+n added up infinitely many times,which is meaningless.

To resolve this problem, we need to redefine our operators PN so as to makethem fit into the completion Uκ(sl2). There is an obvious way to do this: wejust switch by hand the order in the second summation in (2.1.5) to complywith the requirements: ∑

n+m=N ;n≥0

emfn +∑

n+m=N ;n<0

fnem. (2.1.6)

Note, however, that this is not the only way to modify the definition. Anotherway to do it is ∑

n+m=N ;m<0

emfn +∑

n+m=N ;m≥0

fnem. (2.1.7)

Since n and m are constrained by the equation n +m = N , it is easy to seethat this expression also belongs to Uκ(sl2). But it is different from (2.1.6)

2.1 The center 37

because the order of finitely many terms is switched. For example, if N ≥ 0,the terms emfn, 0 ≤ m ≤ N , in (2.1.6) are replaced by fnem in (2.1.7). Sothe difference between the two expressions is (N +1)hn+m, which is of coursea well-defined element of the enveloping algebra.

Thus, the upshot is that there are several inequivalent ways to “regularize”the meaningless expression (2.1.5), which differ at finitely many places. Inwhat follows we will use the second scenario. To write it down in a moreconvenient way, we introduce the notion of normal ordering. For any A,B ∈sl2 (or an arbitrary simple Lie algebra g) we will set

:AmBn: def=

AmBn, m < 0,

BnAm, m ≥ 0.

Then (2.1.7) may be rewritten as∑n+m=N

:emfn: ,

which is the z−N−2 coefficient of :e(z)f(z):, where we apply the normalordering by linearity.

Now we apply the normal ordering to the formal power series P (z) givenby formula (2.1.3). The result is another formal power series

S(z) =∑N∈Z

SNz−N−2 =

12

(:e(z)f(z): + :f(z)e(z): +

12:h(z)h(z):

). (2.1.8)

The corresponding coefficients SN are now well-defined elements of the com-pletion Uκ(sl2).

For a general simple Lie algebra, we write

S(z) =∑N∈Z

SNz−N−2 =

12

dim g∑a=1

:Ja(z)Ja(z): . (2.1.9)

Note that S(z) is independent of the choice of the basis Ja. The coefficientsSN ∈ Uκ(g) of S(z) are called the Segal–Sugawara operators. Are theycentral elements of Uκ(g)? To answer this question, we need to compute thecommutators

[Sn, Am] = SnAm −AmSn

in Uκ(g) for all A ∈ g. The elements Sn are central if and only if thesecommutators vanish.

This computation is not an easy task. Let us first give the answer. Let κc

be the critical invariant inner product on g defined by the formula

κc(A,B) = −12

Trg adA adB. (2.1.10)

38 Vertex algebras

Then we have

[Sn, Am] = −κ− κc

κ0nAn+m. (2.1.11)

Since the invariant inner products on a simple Lie algebra g form a one-dimensional vector space, the ratio appearing in this formula is well-defined(recall that κ0 6= 0 by our assumption).

Formula (2.1.11) comes as a surprise. It shows that the Segal–Sugawaraoperators are indeed central for one specific value of κ, but this value is notκ = 0, as one might naively expect, but the critical one, κ = κc! This maybe thought of as a “quantum correction” due to our regularization scheme(the normal ordering). In fact, if we just formally compute the commutatorsbetween the coefficients of the original (unregularized) series (2.1.2) and Am,we will find that they are central elements at κ = 0, as naively expected. Butthese coefficients are not elements of our completion Uκ(g), so they cannotpossibly define central elements in Uκ(g). (They belong to a different com-pletion of Uκ(g), one that does not act on smooth gκ-modules and hence isirrelevant for our purposes.) The regularized elements Sn become central onlyafter we shift the level by κc. This is the first indication of the special rolethat the critical level κc plays in representation theory of affine Kac–Moodyalgebras.

How does one prove formula (2.1.11)? A direct calculation is tedious andnot very enlightening. Even if we do make it, the next question will be tocompute the commutation relations between the Sn’s (this is related to thePoisson structure on the center, as we will see below), which is a still hardercalculation if we approach it with “bare hands.” This suggests that we needto develop some more serious tools in order to perform calculations of thissort. After all, we are now only discussing the quadratic Casimir element.But what about higher-order central elements?

The necessary tools are provided in the theory of vertex algebras, whichin particular gives us nice and compact formulas for computing the commu-tation relations such as (2.1.11). The idea, roughly, is that the basic objectsare not the elements Ja

n of gκ and the topological algebra Uκ(g) that theygenerate, but rather the generating series Ja(z) and the vertex algebra thatthey generate. We will take up this theory in the next section.

2.2 Basics of vertex algebras

In this section we give a crash course on the theory of vertex algebras, following[FB], where we refer the reader for more details.

Vertex algebras were originally defined by R. Borcherds [Bo], and the foun-dations of the theory were laid down in [FLM, FHL]. The formalism that wewill use in this book is close to that of [FB, K3], and all results on vertexalgebras presented below are borrowed from these two books. We also note

2.2 Basics of vertex algebras 39

that vertex algebras have geometric counterparts: chiral algebras and fac-torization algebras introduced in [BD2]. The connection between them andvertex algebras is explained in [FB].

2.2.1 Fields

Let R be an algebra over C; a formal power series over R in the variablesz1, . . . , zn is a sum of the form∑

i1,...,in∈ZAi1···inz

i11 · · · zin

n .

The set of all such formal power series is denoted by R[[z±11 , . . . , z±1

n ]].Note carefully the difference between formal power series R[[z±1]] (which

can have arbitrarily large and small powers of z), Taylor power series R[[z]](which have no negative powers of z) and Laurent power series R((z)) (whichhave negative powers of z bounded from below).

What operations can be performed on formal power series? We can certainlyadd them, differentiate them, multiply them by polynomials. However, wecannot multiply them by other formal power series. The reason for this isthat the coefficients of the product will consist of infinite sums, e.g.,(∑

n

Anzn

)(∑m

Bmzm

)=∑

n

zn

∑i+j=n

AiBj

.

Nevertheless, we can multiply two formal power series if the variables theyare in are disjoint, so for example f(z)g(w) makes sense as a formal powerseries in the two variables z and w.

A particularly important example of a field is the formal delta-function.This is denoted by δ(z − w)† and is defined by the formula

δ(z − w) =∑n∈Z

znw−n−1.

It has the following easy to check properties:

(i) A(z)δ(z − w) = A(w)δ(z − w),(ii) (z − w)δ(z − w) = 0,(iii) (z − w)n+1∂n

wδ(z − w) = 0.

The first of these properties tells us that

Resz=0 (A(z)δ(z − w)dz) = A(w),

which is a property we would expect the delta-function to have.

† It is not a function of z − w. This is just notation indicating that the properties of thisformal delta-function correspond closely with properties of the usual delta-function.

40 Vertex algebras

We can make the analogy between C[[z±1]] and distributions more precisein the following way. Given a formal power series A(z) in z, define a linearfunctional (“distribution”) φA on the space of polynomials C[z, z−1] by theformula

φA(f(z)) = ResA(z)f(z)dz.

Recall that the product A(z)f(z) is well-defined, since f(z) is a polynomialand so only finite sums turn up in the product. Conversely, given a distribu-tion φ on C[z, z−1], define a formal power series Aφ by

Aφ(z) =∑n∈Z

φ(zn)z−n−1.

It is easy to see that these two operations are inverse to each other. HenceC[[z±1]] is exactly the space of all distributions on the space of polynomialsC[z, z−1].

We can now think of δ(z−w) as being a formal power series in the variablez with w ∈ C× any non-zero complex number. Under the above identificationit is easy to see that we get a delta-function corresponding to w in the usualsense.

Next we define fields as special types of formal power series. Let V be avector space over C, so EndV is an algebra over C. A field is a formal powerseries in EndV [[z±1]]. We write the field as follows

A(z) =∑n∈Z

Anz−n−1.

The power of z chosen is one that will later make much of the notation simpler.Fields must satisfy the following additional property: For each v ∈ V there isan integer N > 0 such that An · v = 0 for all n > N . We may rephrase thiscondition as saying that A(z) · v is a Laurent polynomial for any v ∈ V .

If the vector space V is Z-graded, i.e., V =⊕

n∈Z Vn, then we have theusual concept of homogeneous elements of V as well as homogeneous endo-morphisms: φ ∈ EndV is homogeneous of degree m if φ(Vn) ⊂ Vn+m for alln. In the case of vertex algebras it is common to call the homogeneity degreeof a vector in V the conformal dimension.

2.2.2 Definition

Now we are ready to give the definition of a vertex algebra.A vertex algebra consists of the following data:

(i) A vector space V (the space of states);(ii) A vector in V denoted by |0〉 (the vacuum vector);(iii) An endomorphism T : V → V (the translation operator);


(iv) A linear map Y (·, z) : V → EndV [[z±1]] sending vectors in V to fieldson V (also called vertex operators)

A ∈ V 7→ Y (A, z) =∑n∈Z

A(n)z−n−1.

(the state-field correspondence).

These satisfy the following axioms:

(i) Y (|0〉 , z) = idV ;(ii) Y (A, z) |0〉 = A+ z(. . .) ∈ V [[z]];(iii) [T, Y (A, z)] = ∂zY (A, z);(iv) T |0〉 = 0;(v) (locality) For any two vectors A,B ∈ Z there is a non-negative integer

N such that

(z − w)N [Y (A, z), Y (B,w)] = 0.

It follows from the axioms for a vertex algebra that the action of T may bedefined by the formula

T (A) = A(−2) |0〉 ,

so T is not an independent datum. However, we have included it in the set ofdata, because this makes axioms more transparent and easy to formulate.

A vertex algebra is called Z- (or Z+-) graded if V is a Z- (resp., Z+-) gradedvector space, |0〉 is a vector of degree 0, T is a linear operator of degree 1, andfor A ∈ Vm the field Y (A, z) has conformal dimension m, i.e.,

degA(n) = −n+m− 1.

A particularly simple example of a vertex algebra can be constructed froma commutative associative unital algebra with a derivation.

Let V be a commutative associative unital algebra with a derivation T . Wedefine the vertex algebra structure as follows:

Y (|0〉 , z) = IdV ,

Y (A, z) =∑n>0

zn

n!mult(TnA) = mult(eTzA), (2.2.1)

T = T. (2.2.2)

Here the operators mult(A) in the power series are left multiplication by A.It is an easy exercise to check that this is a vertex algebra structure. The

vertex algebra structure is particularly simple because the axiom of localityhas become a form of commutativity

[Y (A, z), Y (B,w)] = 0.

This is a very special property. Any vertex algebra with this property is called

42 Vertex algebras

commutative. Another property that this vertex algebra structure has thatis unusual is that the formal power series that occur have only non-negativepowers of z. It turns out that these two properties are equivalent.

Lemma 2.2.1 A vertex algebra is commutative if and only if Y (A, z) ∈EndV [[z]] for all A ∈ V .

Proof. If V is commutative then

Y (A, z)Y (B,w) |0〉 = Y (B,w)Y (A, z) |0〉 .

Expanding these in powers of w and taking the constant coefficient in w usingaxiom (ii), we see that Y (A, z)B ∈ V [[z]] for any A and B. This shows thatY (A, z) ∈ EndV [[z]].

Conversely, if Y (A, z) ∈ EndV [[z]] for all A ∈ V , then Y (A, z)Y (B,w) ∈EndV [[z, w]]. Locality then says that

(z − w)NY (A, z)Y (B,w) = (z − w)NY (B,w)Y (A, z).

As (z − w)N has no divisors of zero in EndV [[z, w]], it follows that V iscommutative.

So we have seen that commutative associative unital algebra with a deriva-tion gives rise to a commutative vertex algebra. It is easy to see that the aboveconstruction can be run in the other direction and so these two categories areequivalent. In particular, Z-graded commutative vertex algebras correspondto Z-graded commutative associative algebras with a derivation of degree 1.

2.2.3 More on locality

We have just seen that the property of commutativity in a vertex algebra isvery restrictive. The point of the theory of vertex algebras is that we replace itby a more general axiom; namely, locality. In this sense, one may think of thenotion of vertex algebra as generalizing the familiar notion of commutativealgebra. In this section we will look closely at the locality axiom and try togain some insights into its meaning.

Let v ∈ V be a vector in V and φ : V → C a linear functional on V . GivenA,B ∈ V we can form two formal power series

〈φ, Y (A, z)Y (B,w)v〉 and 〈φ, Y (B,w)Y (A, z)v〉

These two formal power series, which are a priori elements of C[[z±1, w±1]], ac-tually belong to the subspaces C((z))((w)) and C((w))((z)), respectively. Thesetwo subspaces are different: the first consists of bounded below powers of w,but powers of z are not uniformly bounded, whereas the second consists ofbounded below powers of z but not uniformly bounded powers of w.


The intersection of the two spaces consists of those series in z and w whichhave bounded below powers in both z and w. In other words, we have

C((z))((w)) ∩ C((w))((z)) = C[[z, w]][z−1, w−1]. (2.2.3)

Note that C((z))((w)) and C((w))((z)) are closed under multiplication and areactually fields (here we use the terminology “field” in the usual sense!). Theirintersection is a subalgebra C[[z, w]][z−1, w−1]. Therefore within each of thetwo fields we have the fraction field of C[[z, w]][z−1, w−1]. This fraction fieldis denoted by C((z, w)) and consists of ratios f(z, w)/g(z, w), where f, g are inC[[z, w]].

However, the embeddings C((z, w)) into C((z))((w)) and C((w))((z)) are dif-ferent. These embeddings are easy to describe; we simply take Laurent powerseries expansions assuming one of the variables is “small.” We will illustratehow this works using the element 1

z−w ∈ C((z, w)).Assume that w is the “small” variable, and so |w| < |z|. Then we can

expand1

z − w=

1z(1− w

z )= z−1

∑n>0

(wz

)n

in positive powers of w/z. Note that the result will have bounded belowpowers of w and so will lie in C((z))((w)).

Assume now that z is the “small” variable, and so |z| < |w|. We can thenexpand

1z − w

= − 1w(1− z

w )= −z−1

∑n<0

(wz

)n

in negative powers of w/z because |z| < |w|. Note that the result will havebounded below powers of z and so will lie in C((w))((z)).

It is instructive to think of the two rings C((w))((z)) and C((z))((w)) as rep-resenting functions in two variables, which have one of their variables muchsmaller than the other. The “domains of definition” of these functions are|w| |z| and |z| |w|, respectively.

So we have now seen that elements in C((w))((z)) and C((z))((w)), althoughthey can look very different, may in fact be representing the same elementof C((z, w)). This is very similar to the idea of analytic continuation fromcomplex analysis. When these two different elements come from the samerational function in z and w we could think of them as “representing thesame function” (we could even think of the rational function as being thefundamental object rather than the individual representations).

What locality is telling us precisely that the formal power series

〈φ, Y (A, z)Y (B,w)v〉 and 〈φ, Y (B,w)Y (A, z)v〉 (2.2.4)

represent the same rational function in z and w in C((w))((z)) and C((z))((w)),respectively.

44 Vertex algebras

Indeed, the locality axiom states that

(z − w)N 〈φ, Y (A, z)Y (B,w)v〉 and (z − w)N 〈φ, Y (B,w)Y (A, z)v〉

are equal to each other, as elements of C[[z±1, w±1]]. Due to the equality(2.2.3), we find that both of them actually belong to C[[z, w]][z−1, w−1]. So,the formal power series (2.2.4) are representations of the same element ofC[[z, w]][z−1, w−1, (z−w)−1] in C((w))((z)) and C((z))((w)), respectively. If weask in addition that as v and φ vary there is a universal bound on the powerof (z − w) that can occur in the denominator, then we obtain an equivalentform of the locality axiom. This is a reformulation that will be useful in whatfollows.†

2.2.4 Vertex algebra associated to gκ

We now present our main example of a non-commutative vertex algebra, basedon the affine Kac–Moody algebra gκ. Many of the interesting properties ofvertex algebras will be visible in this example.

Let g be a finite-dimensional complex simple Lie algebra with an ordered ba-sis Ja, where a = 1, . . . ,dim g (see Section A.3). Recall that the affine Kac–Moody algebra g has a basis consisting of the elements Ja

n , a = 1, . . . ,dim g, n ∈Z, and 1.

Previously, we grouped the elements associated to Ja into a formal powerseries

Ja(z) =∑n∈Z

Janz−n−1.

This gives us a hint about what some of the fields in this vertex algebra shouldbe.

We should first describe the vector space V on which the vertex algebra isbuilt. We know that we will need a special vector |0〉 in V to be the vacuumvector. We also know that if Ja(z) are indeed vertex operators, then the Ja

n ’sshould be linear operators on V and, by axiom (i), the elements Ja

n for n > 0should annihilate the vacuum vector |0〉.

Notice that the set of Lie algebra elements which are supposed to annihi-late |0〉 form the Lie subalgebra g[[t]] of gκ. Thus, C|0〉 is the trivial one-dimensional representation of g[[t]]. We also define an action of the centralelement 1 on |0〉 as follows: 1|0〉 = 1. Let us denote the resulting represen-tation of g[[t]] ⊕ C1 by Cκ. We can now define a gκ-module by using the

† It may seem slightly strange that we only allow three types of singularities: at z =0, w = 0, and z = w. But these are the only equations that do not depend on thechoice of coordinate, which is a valuable property for us as we will need a coordinate-freedescription of vertex operators and algebras.


induction functor:

Vκ(g) = Indbgκ

g[[t]]⊕C1Cκ = U(gκ) ⊗U(g[[t]]⊕C1)

Cκ.

It is called the vacuum Verma module of level κ.Recall that κ is unique up to a scalar. Therefore it is often convenient

to fix a particular invariant inner product κ0 and write an arbitrary one asκ = kκ0, k ∈ C. This is the point of view taken, for example, in [FB], whereas κ0 we take the inner product with respect to which the squared length ofthe maximal root is equal to 2, and denote the corresponding vacuum moduleby Vk(g).

The structure of the module Vκ(g) is easy to describe. By the Poincare–Birkhoff–Witt theorem, Vκ(g) is isomorphic to U(g⊗ t−1C[t−1])|0〉. Thereforeit has a basis of lexicographically ordered monomials of the form

Ja1n1. . . Jam

nm|0〉, (2.2.5)

where n1 ≤ n2 ≤ . . . ≤ nm < 0, and if ni = ni+1, then ai ≤ ai+1. We define aZ-grading on gκ and on Vκ(g) by the formula deg Ja

n = −n,deg |0〉 = 0. Thehomogeneous graded components of Vκ(g) are finite-dimensional and they arenon-zero only in non-negative degrees. Here is the picture of the first fewhomogeneous components of Vκ(g):

u|0〉

AAAAAAAAAAAAAAA

uJa−1|0〉

u Ja−1J

b−1|0〉uJa

−2|0〉

uJa−3|0〉 u. . . u. . .

The action of gκ is described as follows. The action of Jan with n < 0 is just

the obvious action on U(g ⊗ t−1C[t−1]) ' Vκ(g). To apply Jan with n > 0,

we use the commutation relations in the Lie algebra gκ to move this termthrough to the vacuum vector, which the Ja

n ’s with n > 0 annihilate.Let us illustrate the structure of Vκ(g) in the case of the affine Lie algebra

46 Vertex algebras

sl2. The elements Ja are now denoted by e, f, h and the commutation relationsbetween them are

[h, e] = 2e, [h, f ] = −2f, [e, f ] = h.

For example, consider the vector

e−1f−2 |0〉 ∈ Vκ(sl2).

If we apply h1 to it, we obtain

h1e−1f−2 |0〉 = ([h1, e−1] + e−1h1) f−2 |0〉= (2e0 + e−1h1) f−2 |0〉= 2 ([e0, f−2] + f−2e0) |0〉+ e−1 ([h1, f−2] + f−2h1) |0〉= 2h−2 |0〉+ 0− 2e−1f−1 |0〉+ 0

= 2 (h−2 − e−1f−1) |0〉 .

So, at each step we are simply moving the annihilation operators closer andcloser to the vacuum until we have made them all disappear.

We now have the vector space V = Vκ(g) and vacuum vector |0〉 for ourvertex algebra. We still need to define the translation operator T and thestate-field correspondence Y (·, z).

The translation operator T is defined by interpreting it as the vector field−∂t (the reason for this will become more clear later on). This vector fieldnaturally acts on the Lie algebra g((t)) and preserves the Lie subalgebra g[[t]].Therefore it acts on Vκ(g). Concretely, this means that we have the commu-tation relations

[T, Jan ] = −nJa

n−1,

and the vacuum vector is invariant under T ,

T |0〉 = 0,

as required by axiom (iv) of vertex algebras. These two conditions uniquelyspecify the action of T on the vector space Vκ(g).

2.2.5 Defining vertex operators

Finally, we need to define the state-field correspondence. For the vacuumvector this is determined by axiom (i):

Y (|0〉 , z) = Id .

The elements of the next degree, namely 1, are of the form Ja−1 |0〉. To guess

the form of the vertex operators Y (Ja−1 |0〉 , z) corresponding to them, we first

look at the associated graded space of Vκ(g), which is a commutative vertexalgebra.


To describe this associated graded space, we observe that the Poincare–Bikhoff–Witt filtration on U(gκ) induces one on Vκ(g). The ith term of thisfiltration, which we denote by Vκ(g)≤i, is the span of all monomials (2.2.5)with m ≤ i. The associated graded algebra grVκ(g) with respect to this filtra-tion is the symmetric algebra with generators corresponding to Ja

n with n < 0.To distinguish them from the actual Ja

n ’s, we will denote these generators byJ

a

n.Recall that for any Lie algebra g, the associated graded grU(g) is isomorphic

to Sym g. This implies that for any κ we have

grVκ(g) = Sym(g((t))/g[[t]]) ' Sym(t−1g[[t−1]]).

Thus, grVκ(g) is a commutative unital algebra with a derivation T corre-sponding to the vector field −∂t. It is uniquely determined by the formulaT · Ja

n = −nJa

n−1. Therefore, according to the discussion of Section 2.2.2,grVκ(g) has a natural structure of a commutative vertex algebra. By defini-tion (see formula (2.2.1)), in this vertex algebra we have

Y (Ja

−1, z) =∑n>0

zn

n!mult(Tn · Ja

−1) =∑n>0

mult(Ja

−n−1)zn.

By abusing notation, we will write this as

Y (Ja

−1, z) =∑m<0

Ja

mz−m−1 (2.2.6)

with the understanding that on the right hand side Ja

m stands for the corre-sponding operator of multiplication acting on grVκ(g).

In formula (2.2.6) only “half” of the generators of gκ is involved; namely,those with m < 0. This ensures that the resulting sum has no negative powersof z, as expected in a commutative vertex algebra. Now we generalize thisformula to the case of a non-commutative vertex algebra Vκ(g), in which weare allowed to have negative powers of z appearing in the vertex operators.This leads us to the following proposal for the vertex operator correspondingto Ja

−1 |0〉 ∈ Vκ(g):†

Y (Ja−1 |0〉 , z) =

∑n∈Z

Janz−n−1 = Ja(z).

It is easy to see that these vertex operators satisfy the relations

Y (Ja−1 |0〉 , z) |0〉 = Ja

−1 |0〉+ z(. . .),[T, Y (Ja

−1 |0〉 , z)]

= ∂zY (Ja−1 |0〉 , z)

required by the axioms of vertex algebras.We should also check that these vertex operators satisfy the locality axiom.

To do this we use the commutation relations

[Jan , J

bm] = [Ja, Jb]n+m + nκ(Ja, Jb)δn,−m1 (2.2.7)

† note that Jan refers here to the operator of action of Ja

n on the representation Vκ(g)

48 Vertex algebras

in gκ to evaluate the commutator of Ja(z) and Jb(w):[Ja(z), Jb(w)

]=[Ja, Jb

](w)δ(z − w) + κ(Ja, Jb)∂wδ(z − w).

Now, recalling from (2.2.1) that (z − w)2 annihilates both δ(z − w) and itsderivative, we see that the locality axiom is satisfied.

We now need to define the vertex operators corresponding to the more gen-eral elements of Vκ(g). In fact, we will see in Theorem 2.2.5 below that thedata that we have already defined: |0〉, T and Y (Ja

−1 |0〉 , z) uniquely deter-mine the entire vertex algebra structure on Vκ(g) (provided that it exists!).The reason is that the vectors Ja

−1 |0〉 generate Vκ(g) in the following sense:Vκ(g) is spanned by the vectors obtained by successively applying the coeffi-cients of the vertex operators Y (Ja

−1 |0〉 , z) to the vacuum vector |0〉.Here we will motivate the remaining structure from that on the associated

graded algebra grVκ(g). First of all, we find by an explicit calculation that ingrVκ(g) we have

Y (Ja

n, z) =1

(−n− 1)!∂−n−1

z

∑m<0

Ja

mz−m−1.

This motivates the formula

Y (Jan |0〉 , z) =

1(−n− 1)!

∂−n−1z Ja(z)

in Vκ(g).Next, observe that in any commutative vertex algebra V we have the fol-

lowing simple identity

Y (AB, z) = Y (A, z)Y (B, z),

which follows from formula (2.2.1) and the Leibniz rule for the derivation T

(here on the left hand side AB stands for the ordinary product with respectto the ordinary commutative algebra structure).

Therefore it is tempting to set, for example,

Y (Ja−1J

b−a|0〉, z) = Y (Ja

−1, z)Y (Jb−1, z) = Ja(z)Jb(z).

However, we already know that the coefficients of the product on the righthand side are not well-defined as endomorphisms of Vκ(g). The problem isthat the annihilation operators do not appear to the right of the creationoperators. We had also suggested a cure: switching the order of some of theterms.

We will now define this procedure, called normal ordering, in a more sys-tematic way. Let us define, for a formal power series

f(z) =∑n∈Z

fnzn ∈ R[[z±1]],


where R is any C-algebra, the series f(z)+ to be the part with non-negativepowers of z and f(z)− to be the part with strictly negative powers of z:

f+(z) =∑n≥0

fnzn, f−(z) =

∑n<0

fnzn.

The normally ordered product of two fields A(z) and B(z) is then definedto be

:A(z)B(z): def= A(z)+B(z) +B(z)A(z)−.

It is an easy exercise (left for the reader) to check that :A(z)B(z): is againa field (although the coefficients of z may be infinite sums, when applied toa vector they become finite sums). When we have more than two fields, thenormally ordered product is defined from right to left, e.g.,

:A(z)B(z)C(z): = :A(z) (:B(z)C(z):) : .

This formula looks somewhat ad hoc, but we will see in Theorem 2.2.5 belowthat it is uniquely determined by the axioms of vertex algebra.

Another way to write the definition for normally ordered product is to useresidues. For a function in two variables F (z, w) define F|z|>|w| to be theexpansion assuming z is the “large” variable (and similarly for F|w|>|z|). Forexample(

1z − w

)|z|>|w|

= z−1∞∑

n=0

(wz

)n

,

(1

z − w

)|w|>|z|

= −w−1∞∑

n=0

( zw

)n

.

Then we can represent the normally ordered product as

:A(z)B(z): =

Resw=0

((1

w − z

)|w|>|z|

A(w)B(z)−(

1w − z

)|z|>|w|

B(z)A(w)

)dw.

This identity is easy to see from the following formulas

Resz=0

(A(z)

(1

z − w

)|z|>|w|

)dz = A(w)+,

Resz=0

(A(z)

(1

z − w

)|w|>|z|

)dz = −A(w)−,

and these may be proved by simple calculation.Using the normal ordering by induction, we arrive at the following guess

for a general vertex operator in Vκ(g):

Y (Ja1n1. . . Jam

nm|0〉 , z) =

1(−n1 − 1)!

. . .1

(−nm − 1)!:∂−n1−1

z Ja1(z) . . . ∂−nm−1z Jam(z): . (2.2.8)

50 Vertex algebras

Theorem 2.2.2 The above formulas define the structure of a Z+-graded vertexalgebra on Vκ(g).

2.2.6 Proof of the Theorem

We have Y (|0〉 , z) = Id by definition. Next, we need to check

Y (A, z) |0〉 = A+ z(. . .)

to see that the vacuum axiom holds. This is clearly true when A = |0〉. Wethen prove it in general by induction: assuming it to hold for Y (B, z), whereB ∈ Vκ(g)≤i, we find for any A ∈ g that

Y (A−nB |0〉 , z) |0〉 =1

(−n− 1)!:∂−n−1

z A(z) · Y (B, z): |0〉 = A−nB + z(. . .).

The translation axiom (iii) boils down to proving the identity

[T, :A(z)B(z):] = ∂z:A(z)B(z): ,

assuming that [T,A(z)] = ∂zA(z) and [T,B(z)] = ∂zB(z). We leave this tothe reader, as well as the fact that the Z+-grading is compatible with thevertex operators.

The locality follows from Dong’s lemma presented below, which shows thatnormally ordered products of local fields remain local. It is also clear thatderivatives of local fields are still local. So, all we need to do is check that thegenerating fields Ja(z) are mutually local. From the commutation relationswe obtain that[

Ja(z), Jb(w)]

=[Ja, Jb

](w)δ(z − w)− κ(Ja, Jb)∂wδ(z − w).

From this it is clear that

(z − w)2[Ja(z), Jb(w)

]= 0,

and so we have the required locality.This completes the proof modulo the following lemma.

Lemma 2.2.3 (Dong) If A(z), B(z), C(z) are mutually local fields, then: A(z)B(z) : and C(z) are mutually local as well.

Proof. The result will follow (by taking Resx=0) if we can show that thefollowing two expressions are equal after multiplying by a suitable power of(y − z):

F =(

1x− y

)|x|>|y|

A(x)B(y)C(z)−(

1x− y

)|y|>|x|

B(y)A(x)C(z),

G =(

1x− y

)|x|>|y|

C(z)A(x)B(y)−(

1x− y

)|y|>|x|

C(z)B(y)A(x).


As A, B and C are mutually local, we know that there is an integer N suchthat

(x− y)NA(x)B(y) = (x− y)NB(y)A(x)

(y − z)NB(y)C(z) = (y − z)NC(z)B(y)

(x− z)NA(x)C(z) = (x− z)NC(z)A(x).

We will now show that

(y − z)3NF = (y − z)3NG.

The binomial identity gives

(y − z)3N =2N∑n=0

(2Nn

)(y − x)2N−n(x− z)n(y − z)N .

Now, if 0 6 n < N the power of (y−x) is large enough that we can swap A(x)and B(y); the two terms in F (and G) then cancel, so these do not contribute.For n ≥ N the powers of (x − z) and (y − z) are large enough that we canswap A(x), C(z) as well as B(y), C(z). This allows us to make terms in F

the same as those in G. Hence

(y − z)3NF = (y − z)3NG,

and we are done.

The proof of Theorem 2.2.2 works in more generality than simply the caseof an affine Kac–Moody algebra. In this form it is called the (weak) recon-struction theorem.

Theorem 2.2.4 (Weak Reconstruction) Let V be a vector space, |0〉 avector of V , and T an endomorphism of V . Let

aα(z) =∑n∈Z

aα(n)z

−n−1,

where α runs over an ordered set I, be a collection of fields on V such that

(i) [T, aα(z)] = ∂zaα(z);

(ii) T |0〉 = 0, aα(z) |0〉 = aα + z(. . .);(iii) aα(z) and aβ(z) are mutually local;(iv) the lexicographically ordered monomials aα1

(n1). . . aαm

(nm) |0〉 with ni < 0form a basis of V .

Then the formula

Y (aα1(n1)· · · aαm

(nm) |0〉 , z) =

1(−n1 − 1)!

. . .1

(−nm − 1)!:∂−n1−1aα1(z) · · · ∂−nm−1aαn(z): , (2.2.9)

52 Vertex algebras

where ni < 0, defines a vertex algebra structure on V such that |0〉 is thevacuum vector, T is the translation operator and Y (aα, z) = aα(z) for allα ∈ I.

To prove this we simply repeat the proof used for the case of V = Vκ(g).The reconstruction theorem is actually true in a much less restrictive case:

we do not have to assume that the vectors in (iv) form a basis, merely thatthey span V . And even in this case it is possible to derive that the resultingvertex algebra structure is unique.

Theorem 2.2.5 (Strong Reconstruction)

Let V be a vector space, equipped with the structures of Theorem 2.2.4satisfying conditions (i)–(iii) and the condition

(iv’) The vectors aα1(−j1−1) · · · a

αn

(−jn−1) |0〉 with js > 0 span V . Then thesestructures together with formula (2.2.9) define a vertex algebra structure onV .

Moreover, this is the unique vertex algebra structure on V satisfying condi-tions (i)–(iii) and (iv’) and such that Y (aα, z) = aα(z) for all α ∈ I.

We will not prove this result here, referring the reader to [FB], Theorem4.4.1. In particular, we see that there there was no arbitrariness in our as-signment of vertex operators in Section 2.2.5.

2.3 Associativity in vertex algebras

We have now made it about half way towards our goal of finding a properformalism for the Segal–Sugawara operators Sn defined by formula (2.1.9).We have introduced the notion of a vertex algebra and have seen that vertexalgebras neatly encode the structures related to the formal power series suchas the generating series S(z) of the Segal–Sugawara operators. In fact, theseries S(z) itself is one of the vertex operators in the vertex algebra Vκ(g):

S(z) =12

∑a=1

Y (Ja−1Ja,−1|0〉, z).

What we need to do now is to learn how to relate properties of the vertexoperators such as S(z), emerging from the vertex algebra structure, and prop-erties of their Fourier coefficients Sn, which are relevant to the description ofthe center of the completed enveloping algebra of gκ that we are interestedin. In order to do that, we need to develop the theory of vertex algebras abit further, and in particular understand the meaning of associativity in thiscontext.

2.3 Associativity in vertex algebras 53

2.3.1 Three domains

We have already seen that locality axiom for a vertex algebra is telling us thatthe two formal power series

Y (A, z)Y (B,w)C and Y (B,w)Y (A, z)C

are expansions of the same element from V [[z, w]][z−1, w−1, (z−w)−1] in twodifferent domains. One of these expansions, V ((z))((w)), corresponds to w

being “small;” the other, V ((w))((z)), corresponds to z being “small.” If wethink of the points z and w as being complex numbers (in other words, pointson the Riemann sphere), we can think of these two expansions as being donein the domains “w is very close to 0” and “w is very close to ∞.” There isnow an obvious third choice: “w is very close to z,” which we have not yetdiscussed.

Algebraically, the space corresponding to the domain “w is very close to z”is V ((w))((z−w)) (or alternatively, V ((z))((z−w)); these are actually identical).The expression, in terms of vertex algebras, that we expect to live in this spaceis

Y (Y (A, z − w)B,w)C.

To see this, we look at the case of commutative vertex algebras. In a commu-tative vertex algebra, locality is equivalent to commutativity,

Y (A, z)Y (B,w)C = Y (B,w)Y (A, z)C,

but we also have the identity

Y (A, z)Y (B,w)C = Y (Y (A, z − w)B,w)C

which expresses the associativity property of the underlying commutative al-gebra. In a general vertex algebra, locality holds only in the sense of “ana-lytic continuation” from different domains. Likewise, we should expect thatthe associativity property also holds in a similar sense. More precisely, weexpect that Y (Y (A, z − w)B,w)C is the expansion of the same element ofV [[z, w]][z−1, w−1, (z−w)−1], as for the other two expressions, but now in thespace V ((w))((z −w)) (i.e., assuming z −w to be “small”). We will now showthat this is indeed the case.

2.3.2 Some useful facts

We start with a couple of basic but useful results about vertex algebras.

Lemma 2.3.1 Suppose that U is a vector space, f(z) ∈ U [[z]] is a powerseries, and R ∈ EndU is an endomorphism. If f(z) satisfies

∂zf(z) = Rf(z),

54 Vertex algebras

then it is uniquely determined by the value of f(0).

Proof. A simple induction shows that f(z) must be of the form

f(z) = K +R(K)z +R2

2!(K)z2 +

R3

3!(K)z3 + . . . ,

so it is determined by K = f(0).

Corollary 2.3.1 For any vector A in a vertex algebra V we have

Y (A, z) |0〉 = ezTA.

Proof. Both sides belong to V [[z]]. Therefore, in view of Lemma 2.3.1,it suffices to show that they satisfy the same differential equation, as theirconstant terms are both equal to A. It follows from the vertex algebra axiomsthat

∂zY (A, z)|0〉 = [T, Y (A, z)]|0〉 = TY (A, z)|0〉.

On the other hand, we obviously have ∂zezTA = TezTA.

Lemma 2.3.2 In any vertex algebra we have

ewTY (A, z)e−wT = Y (A, z + w),

where negative powers of (z +w) are expanded as power series assuming thatw is “small” (i.e., in positive powers of w/z).

Proof. In any Lie algebra we have the following identity:

ewTGe−wT =∑n>0

wn

n!(adT )nG

So, in our case, using the formula [T, Y (A, z)] = ∂zY (A, z) and the fact that[T, ∂z] = 0, we find that

ewTY (A, z)e−wT =∑n>0

wn

n!(adT )nY (A, z)

=∑n>0

wn

n!∂n

z Y (A, z)

= ew∂zY (A, z).

To complete the proof, we use the identity ew∂zf(z) = f(z+w) in R[[z±1, w]],which holds for any f(z) ∈ R[[z±1]] and any C-algebra R.

This lemma tells us that exponentiating the infinitesimal translation oper-ator T really does give us a translation operator z 7→ z + w.


Proposition 2.3.2 (Skew Symmetry) In any vertex algebra we have theidentity

Y (A, z)B = ezTY (B,−z)A

in V ((z)).

Proof. By locality, we know that there is a large integer N such that

(z − w)NY (A, z)Y (B,w) |0〉 = (z − w)NY (B,w)Y (A, z) |0〉 .

This is actually an equality in V [[z, w]] (note that there are no negative powersof w on the left and no negative powers of z on the right). Now, by the aboveresults we compute

(z − w)NY (A, z)Y (B,w) |0〉 = (z − w)NY (B,w)Y (A, z) |0〉⇒ (z − w)NY (A, z)ewTB = (z − w)NY (B,w)ezTA

⇒ (z − w)NY (A, z)ewTB = (z − w)NezTY (B,w − z)A⇒ (z)NY (A, z)B = (z)NezTY (B,−z)A⇒ Y (A, z)B = ezTY (B,−z)A

In the fourth line we have set w = 0, which is allowed as there are no negativepowers of w in the above expressions.

In terms of the Fourier coefficients, we can write the skew symmetry prop-erty as

A(n)B = (−1)n+1

(B(n)A− T (B(n−1)A) +

12T 2(B(n−2)A)− . . .

).

2.3.3 Proof of associativity

We now have enough to prove the associativity property.

Theorem 2.3.3 In any vertex algebra the expressions

Y (A, z)Y (B,w)C, Y (B,w)Y (A, z)C, and Y (Y (A, z − w)B,w)C

are the expansions, in

V ((z))((w)), V ((w))((z)), and V ((w))((z − w)),

respectively, of one and the same element of V [[z, w]][z−1, w−1, (z − w)−1].

Proof. We already know this from locality for the first two expressions, sowe only need to show it for the first and the last one. Using Proposition 2.3.2and Lemma 2.3.2, we compute

Y (A, z)Y (B,w)C = Y (A, z)ewTY (C,−w)B

= ewTY (A, z − w)Y (C,−w)B

56 Vertex algebras

(note that it is okay to multiply on the left by the power series ewT as it hasno negative powers of w). In this final expression we expand negative powersof (z −w) assuming that w is “small.” This defines a map V ((z −w))((w))→V ((z))((w)), which is easily seen to be an isomorphism that intertwines theembeddings of V [[z, w]][z−1, w−1, (z − w)−1] into the two spaces.

On the other hand, we compute, again using Proposition 2.3.2,

Y (Y (A, z − w)B,w)C = Y

(∑n

A(n)B(z − w)−n−1, w

)C

=∑

n

Y (A(n)B,w)C(z − w)−n−1

=∑

n

ewTY (C,−w)A(n)B(z − w)−n−1

= ewTY (C,−w)Y (A, z − w)B.

This calculation holds in V ((w))((z − w)).By locality we know that

Y (C,−w)Y (A, z − w)B and Y (A, z − w)Y (C,−w)B

are expansions of the same element of V [[z, w]][z−1, w−1, (z − w)−1]. Thisimplies that Y (Y (A, z−w)B,w)C and Y (A, z)Y (B,w)C are also expansionsof the same element.

We have now established the associativity property of vertex algebras. Itis instructive to think of this property as saying that

Y (A, z)Y (B,w) = Y (Y (A, z − w)B,w) =∑n∈Z

Y (A(n)B,w)(z − w)−n−1. (2.3.1)

This is very useful as it gives a way to represent the product of two vertexoperators as a linear combination of vertex operators. However, we haveto be careful in interpreting this formula. The two sides are formal powerseries in two different vector spaces, V ((z)) and V ((w))((z − w)). They do“converge” when we apply the terms to a fixed vector C ∈ V , but even thenthe expressions are not equal as they are expansions of a common “function”in two different domains.

Written in the above form, and with the above understanding, the equality(2.3.1) is known as the operator product expansion (or OPE). Formulasof this type originally turned up in the physics literature on conformal fieldtheory. One of the motivations for developing the theory of vertex algebraswas to find a mathematically rigorous interpretation of these formulas.


2.3.4 Corollaries of associativity

We now look at consequences of the associativity law. In particular, we willsee that our previous definitions of normally ordered product and the formulafor the vertex operators in the case of Vκ(g) are basically unique.

Lemma 2.3.4 Suppose that φ(z) and ψ(w) are two fields. Then the followingare equivalent:

(i) [φ(z), ψ(w)] =N−1∑i=0

1i!γi(w)∂i

wδ(z − w);

(ii) φ(z)ψ(w) =N−1∑i=0

γi(w)(

1(z − w)i+1

)|z|>|w|

+ :φ(z)ψ(w):

and ψ(w)φ(z) =N−1∑i=0

γi(w)(

1(z − w)i+1

)|w|>|z|

+ :φ(z)ψ(w): ,

where the γi(w) are fields, N is a non-negative integer and

:φ(z)ψ(w): = φ+(z)ψ(w) + ψ(w)φ−(z).

Proof. Assuming (ii) we see the commutator [φ(z), ψ(w)] is the differenceof expansions of (z−w)−i−1 in the domains |z| > |w| and |w| > |z|. It is clearthat (

1z − w

)|z|>|w|

−(

1z − w

)|w|>|z|

= δ(z − w).

Differentiating i times with respect to w, we obtain(1

(z − w)i+1

)|z|>|w|

−(

1(z − w)i+1

)|w|>|z|

=1i!∂i

wδ(z − w).

This gives us (i).Conversely, assume (i) and write

φ(z)ψ(w) = (φ(z)+ + φ(z)−)ψ(w)

= (φ(z)+ψ(w) + ψ(w)φ(z)−) + (φ(z)−ψ(w)− ψ(w)φ(z)−)

= :φ(z)ψ(w): + [φ(z)−, ψ(w)] .

Taking the terms with powers negative in z in the right hand side of (i), weobtain the first formula in (ii). The same calculation works for the productψ(w)φ(z).

Now suppose that the fields φ(z) and ψ(w) are vertex operators Y (A, z) andY (B,w) in a vertex algebra V . Then locality tells us that the commutator[Y (A, z), Y (B,w)] is annihilated by (z−w)N for some N . It is easy to see thatthe kernel of the operator of multiplication by (z−w)N in EndV [[z±1, w±1]] is

58 Vertex algebras

linearly generated by the series of the form γ(w)∂iwδ(z−w), i = 0, . . . , N − 1.

Thus, we can write

[Y (A, z), Y (B,w)] =N−1∑i=0

1i!γi(w)∂i

wδ(z − w),

where γi(w), i = 0, . . . , N − 1, are some fields on V (we do not know yet thatthey are also vertex operators). Therefore we obtain that Y (A, z) and Y (B,w)satisfy condition (i) of Lemma 2.3.4. Hence they also satisfy condition (ii):

Y (A, z)Y (B,w) =N−1∑i=0

γi(w)(z − w)i+1

+ :Y (A, z)Y (B,w): , (2.3.2)

where by (z − w)−1 we understand its expansion in positive powers of w/z.We obtain that for any C ∈ V the series Y (A, z)Y (B,w)C ∈ V ((z))((w)) is

an expansion of(N−1∑i=0

γi(w)(z − w)i+1

+ :Y (A, z)Y (B,w):

)C ∈ V [[z, w]][z−1, w−1, (z − w)−1].

Using the Taylor formula, we obtain that the expansion of this element inV ((w))((z − w)) is equal toN−1∑

i=0

γi(w)(z − w)i+1

+∑m≥0

(z − w)m

m!:∂m

w Y (A,w) · Y (B,w):

C. (2.3.3)

The coefficient in front of (z − w)k, k ∈ Z, in the right hand side of (2.3.3)must be equal to the corresponding term in the right hand side of formula(2.3.1). Let us first look at the terms with k ≥ 0. Comparison of the twoformulas gives

Y (A(n)B, z) =1

(−n− 1)!:∂−n−1

z Y (A, z) · Y (B, z): , n < 0. (2.3.4)

In particular, setting B = |0〉, n = −2 and recalling that A(−2)|0〉 = TA,we find that

Y (TA, z) = ∂zY (A, z). (2.3.5)

Using formula (2.3.4), we obtain the following corollary by induction on m(recall that our convention is that the normal ordering is nested from right toleft):

Corollary 2.3.3 For any A1, . . . , Am ∈ V , and n1, . . . , nm < 0, we have

Y (A1(n1)

. . . Ak(nm)|0〉, z)

=1

(−n1 − 1)!. . .

1(−nm − 1)!

:∂−n1−1z Y (A1, z) · . . . · ∂−nm−1

z Y (Am, z): .


This justifies formulas (2.2.8) and (2.2.9).Now we compare the coefficients in front of (z − w)k, k < 0, in formulas

(2.3.1) and (2.3.3). We find that

γi(w) = Y (A(i)B,w), i ≥ 0,

and so formula (2.3.2) can be rewritten as

Y (A, z)Y (B,w) =∑n≥0

Y (A(n)B,w)(z − w)n+1

+ :Y (A, z)Y (B,w): . (2.3.6)

Note that unlike formula (2.3.1), this identity makes sense in EndV [[z−1, w−1]]if we expand (z − w)−1 in positive powers of w/z.

Now Lemma 2.3.4 implies the following commutation relations:

[Y (A, z), Y (B,w)] =∑n≥0

1n!Y (A(n)B,w)∂n

wδ(z − w). (2.3.7)

Expanding both sides of (2.3.7) as formal power series and using the equality

1n!∂n

wδ(z − w) =∑m∈Z

(m

n

)z−m−1wm−n,

we obtain the following identity for the commutators of Fourier coefficients ofarbitrary vertex operators:

[A(m), B(k)] =∑n≥0

(m

n

)(A(n)B)(m+k−n). (2.3.8)

Here, by definition, for any m ∈ Z,(m

n

)=m(m− 1) . . . (m− n+ 1)

n!, n ∈ Z>0;

(m

0

)= 1.

Several important remarks should be made about this formula. It showsthat the collection of all Fourier coefficients of all vertex operators form aLie algebra; and the commutators of Fourier coefficients depend only on thesingular terms in the OPE.

We will usually simplify the notation in the above lemma, writing

φ(z)ψ(w) =N−1∑i=0

γi(w)(

1(z − w)i+1

)|z|>|w|

+ : φ(z)ψ(w) :

as

φ(z)ψ(w) ∼N−1∑i=0

γi(w)(z − w)i+1

.

In the physics literature this is what is usually written down for an OPE. Whatwe have basically done is to remove all the terms which are non-singular at

60 Vertex algebras

z = w. We can do this because the structure of all the commutation relationsdepends only on the singular terms.

3

Constructing central elements

In the previous chapter we started investigating the center of the completedenveloping algebra Uκ(g) of gκ. After writing some explicit formulas we re-alized that we needed to develop some techniques in order to perform thecalculations. This led us to the concept of vertex algebras and vertex opera-tors. We will now use this formalism in order to describe the center.

In Section 3.1 we will use the general commutation relations (2.3.8) betweenthe Fourier coefficients of vertex operators in order to prove formula (2.1.11)for the commutation relations between the Segal–Sugawara operators and thegenerators of gκ. However, we quickly realize that this proof gives us thecommutation relations between these operators not in Uκ(g), as we wanted,but in the algebra of endomorphisms of Vκ(g). In order to prove that thesame formula holds in Uκ(g), we need to work a little harder. To this endwe associate in Section 3.2 to an arbitrary vertex algebra V a Lie algebraU(V ). We then discuss the relationship between U(Vκ(g)) and Uκ(g). Theconclusion is that formula (2.1.11) does hold in the completed envelopingalgebra Uκ(g). In particular, we obtain that the Segal–Sugawara operatorsare central elements of Uκ(g) when κ = κc, the critical level.

Next, in Section 3.3 we introduce the notion of center of a vertex algebra.We show that the center of the vertex algebra Vκ(g) is a commutative algebrathat is naturally realized as a quotient of the center of Uκ(g). We show thatthe center of Vκ(g) is trivial if κ 6= κc. Our goal is therefore to find the centerz(g) of Vκc

(g). We will see below that this will enable us to describe the centerof Uκ(g) as well. As the first step, we discuss the associated graded analogueof z(g). In order to describe it we need to make a detour to the theory of jetschemes in Section 3.4.

The results of Section 3.4 are already sufficient to find z(g) for g = sl2:we show in Section 3.5 that z(sl2) is generated by the Segal–Sugawara oper-ators. However, we are not satisfied with this result as it does not give us acoordinate-independent description of the center. Therefore we need to findhow the group of changes of coordinates on the disc acts on z(sl2). As theresult of this computation, performed in Section 3.5, we find that z(sl2) is

61

62 Constructing central elements

canonically isomorphic to the algebra of functions on the space of projectiveconnections on the disc. This is the prototype for the description of the centerz(g) obtained in the next chapter, which is one of the central results of thisbook.

3.1 Segal–Sugawara operators

Armed with the technique developed in the previous chapter, we are nowready to prove formula (2.1.11) for the commutation relations between theSegal–Sugawara operators Sn and the generators Ja

n of gκ.

3.1.1 Commutation relations with gκ

We have already found that the commutation relations (2.2.7) between theJa

n ’s may be recorded nicely as the commutation relations for the fields Ja(z):[Ja(z), Jb(w)

]=[Ja, Jb

](w)δ(z − w) + κ(Ja, Jb)∂wδ(z − w).

Using the formalism of the previous section, we rewrite this in the equivalentform of the OPE

Ja(z)Jb(w) ∼[Ja, Jb

](w)

(z − w)+κ(Ja, Jb)(z − w)2

.

Now recall that

S(z) =12

∑a

Y (Ja−1Ja,−1|0〉, z) =

12

∑a

:Ja(z)Ja(z):

where Ja is the dual basis to Ja under some fixed inner product κ0. Weknow from our previous discussions about normally ordered products thatthis is a well-defined field, so we would now like to work out the commutationrelations of the Fourier coefficients of S(z) and the Ja

(n)’s, which are the Fouriercoefficients of Ja(z).

Let

S =12

∑a

Ja(−1)Ja,(−1) |0〉 , (3.1.1)

so that S is the element of Vκ(g) generating the field S(z):

S(z) = Y (S, z) =∑n∈Z

S(n)z−n−1.

Note that since we defined the operators Sn by the formula

S(z) =∑n∈Z

Snz−n−2,

we have

Sn = S(n+1).

3.1 Segal–Sugawara operators 63

We need to compute the OPE

Y (Jb, z)Y (S,w) ∼∑n>0

Y (Jb(n)S,w)

(z − w)n+1.

This means we need to compute the elements

12Jb

(n)

∑a

Ja(−1)Ja,(−1) |0〉 ∈ Vκ(g), n ≥ 0.

As we will see, this is much easier than trying to compute the commutatorsof the infinite sum representing Sn with Jb

m.The vertex algebra Vκ(g) is Z+-graded, and the degree of S is equal to

2. Therefore we must have n 6 2, for otherwise the resulting element is ofnegative degree, and there are no such non-zero elements. Thus, we havethree cases to consider:

n=2. In this case, repeatedly using the commutation relations and the in-variance of the inner product κ quickly shows that the result is

12

∑a

κ(Jb, [Ja, Ja]

)|0〉 .

If we now choose Ja to be an orthonormal basis, which we may dowithout loss of generality, so that Ja = Ja, we see that this is 0.

n=1. We obtain ∑a

κ(Jb, Ja)Ja−1 |0〉+

12

∑a

[[Jb, Ja], Ja]−1 |0〉 .

The first of these terms looks like the decomposition of Jb−1 using the

basis Ja−1. However, we have to be careful that κ is not necessarily

the same inner product as κ0; they are multiples of each other. So,the first term is just κ

κ0Jb−1 |0〉.

The second term is the action of the Casimir element of U(g) on theadjoint representation of g, realized as a subspace g⊗ t−1|0〉 ⊂ Vκ(g).This is a central element and so it acts as a scalar, λ, say. Anyfinite-dimensional representation V of g gives rise to a natural innerproduct by

κV (x, y) = Tr ρV (x)ρV (y).

In particular κg is the inner product from the adjoint representa-tion known as the Killing form which we have seen before in Sec-tion 1.3.5. If we pick κ0 = κg, it is obvious that the scalar λ shouldbe 1/2 (due to the overall factor 1/2 in the formula). So, for a general

κ0 the second term is12κg

κ0Jb−1 |0〉.


n=0. We get12

∑a

(Ja−1[J

b, Ja]−1 + [Jb, Ja]−1Ja,−1

).

This corresponds to taking the commutator with the Casimir elementand hence is 0 (as the Casimir is central).

Hence the OPE is

Jb(z)S(w) ∼κ+ 1

2κg

κ0

Jb(w)(z − w)2

.

So, we find that there is a critical value for the inner product κ, forwhich these two fields, S(z) and Jb(z) (and hence their Fourier coefficients),commute with each other. The critical value is κc = − 1

2κg.Using formula (2.3.8) and the first form of the OPE, we immediately obtain

the commutation relations

[Jan , S(m)] =

κ− κc

κ0nJa

n+m−1.

Recall that we define Sn to be S(n+1). In other words, we shift the grading by1, so that the degree of Sn becomes −n (but note that we have Ja

n = Ja(n)).

Then we obtain the following formula, which coincides with (2.1.11):

[Sn, Jam] = −κ− κc

κ0nJa

n+m.

Thus, we find that when κ = κc the operators Sn commute with the actionof the Lie algebra g, which is what we wanted to show.

We remark that the ratio 12

κg

κ0is related to the dual Coxeter number of

g. More precisely, if κ0 is chosen so that the maximal root has squared length2, which is the standard normalization [K2], then this ratio is equal to h∨,the dual Coxeter number of g (for example, it is equal to n for g = sln). Forthis choice of κ0 the above formula becomes

[Sn, Jam] = −(k + h∨)nJa

n+m.

Even away from the critical value the commutation relations are quite nice.To simplify our formulas, for κ 6= κc we rescale S and S(w) by setting

S =κ0

κ+ 12κg

S, S(z) =κ0

κ+ 12κg

S(z).

Then we have the OPE

Ja(z)S(w) ∼ Ja(w)(z − w)2

.

Note that often this is written in the opposite order, which can be achievedby using locality, swapping z and w and using Taylor’s formula:

S(z)Ja(w) ∼ Ja(w)(z − w)2

+∂wJ

a(w)(z − w)

. (3.1.2)


Using formula (2.3.8) and the first form of the OPE, we find that

[Jan , S(m)] = nJa

n+m−1.

Let us set Sn = S(n+1). Then the commutation relations may be rewritten asfollows:

[Sn, Jam] = −mJa

n+m.

These commutation relations are very suggestive. Recall that Jan stands for

the element Ja ⊗ tn in gκ. Thus, the operations given by ad(Sn) look verymuch like

ad(Sn) = −tn+1∂t.

One can now ask if the operators ad(Sn) actually generate the Lie algebra ofvector fields of this form. It will turn out that this is almost true (“almost”because we will actually obtain a central extension of this Lie algebra).

To see that, we now compute the OPE for S(z)S(w).

3.1.2 Relations between Segal–Sugawara operators

To perform the calculation we need to compute the element of Vκ(g) given bythe formula

S(n)12

κ0

κ− κc

∑a

Ja(−1)Ja,(−1) |0〉

for n ≥ 0. From the grading considerations we find that we must have n 6 3(otherwise the resulting element is of negative degree). So there are four casesto consider:

n=3 In this case, repeatedly using the above commutation relations quicklyshows that the result is

12

κ0

κ− κc

∑a

κ (Ja, Ja) |0〉 .

Hence we obtain 12

κκ−κc

dim g |0〉.n=2 We obtain

12

κ0

κ− κc

∑a

[Ja, Ja](−1) |0〉 ,

which is 0, as we can see by picking Ja = Ja.n=1 We obtain

12

κ0

κ− κc

∑a

2Ja(−1)Ja,(−1) |0〉 ,

which is just 2S.


n=0 We obtain12

κ0

κ− κc

∑a

(Ja

(−1)Ja,(−2) + Ja(−2)Ja,(−1)

)|0〉 ,

which is just T (S).

Thus, we see that the OPE is

S(z)S(w) ∼κ

κ−κcdim g/2

(z − w)4+

2S(w)(z − w)2

+∂wS(w)(z − w)

.

We denote the constant occurring in the first term as cκ/2 (later we will seethat cκ is the so-called central charge). We then use this OPE and formula(2.3.8) to compute the commutation relations among the coefficients Sn =S(n+1):

[Sn, Sm] = (n−m)Sn+m +n3 − n

12cκδn,−m. (3.1.3)

These are the defining relations of the Virasoro algebra, which we discuss inthe next section.

3.1.3 The Virasoro algebra

Let K = C((t)) and consider the Lie algebra

DerK = C((t))∂t

of (continuous) derivations of K with the usual Lie bracket

[f(t)∂t, g(t)∂t] = (f(t)g′(t)− f ′(t)g(t))∂t.

The Virasoro algebra is the central extension of DerK. The central elementis denoted by C and so we have the short exact sequence

0 −→ CC −→ V ir −→ DerK −→ 0.

As a vector space, V ir = DerK⊕CC, so V ir has a topological basis given by

Ln = −tn+1∂t, n ∈ Z,

and C (we include the minus sign in order to follow standard convention).The Lie bracket in the Virasoro algebra is given by the formula

[Ln, Lm] = (n−m)Ln+m +n3 − n

12δn+mC. (3.1.4)

In fact, it is known that V ir is the universal central extension of DerK, justlike gκ is the universal central extension of g((t)).

The relations (3.1.4) become exactly the relations between the operators Sn

found in formula (3.1.3) if we identify C with cκ. This means that the Sn’sgenerate an action of the Virasoro algebra on Vκ(g) for κ 6= κc.


We now construct a vertex algebra associated to the Virasoro algebra in asimilar way to the affine Kac–Moody algebra case.

Let O = C[[t]] and consider the Lie algebra DerO = C[[t]]∂t ⊂ DerK. It istopologically generated by L−1, L0, L1, . . . . The Lie algebra DerO⊕CC hasa one-dimensional representation Cc where DerO acts by 0 and the centralelement C acts as the scalar c ∈ C. Inducing this representation to theVirasoro algebra gives us a V ir-module, which we call the vacuum module ofcentral charge c,

Virc = IndV irDerO⊕CC Cc = U(V ir) ⊗

U(DerO⊕CC)Cc.

This will be the vector space underlying our vertex algebra. By the Poincare–Birkhoff–Witt theorem, a basis for this module is given by the monomials

Ln1Ln2 · · ·Lnm|0〉 ,

where n1 6 n2 6 · · · 6 nm < −1 and |0〉 denotes a generating vector for therepresentation Cc. This will be the vacuum vector of our vertex algebra.

We define a Z-grading on V ir by setting degLn = −n. Setting deg |0〉 = 0,we obtain a Z+-grading on Virc. Note that the subspace of degree 1 is zero,and the subspace of degree 2 is one-dimensional, spanned by L−2|0〉.

Recall that the operator L−1 realizes −∂t, and so we choose it to be thetranslation operator T . We then need to define the state-field correspondence.By analogy with the vertex algebra Vκ(g), we define

Y (L−2 |0〉 , z) = T (z) def=∑n∈Z

Lnz−n−2.

Note that the power of z in front of Ln has to be −n− 2 rather than −n− 1,because we want Virc to be a Z+-graded vertex algebra (in the conventions ofSection 2.2.2). The commutation relations (3.1.4) imply the following formula:

[T (z), T (w)] =C

12∂3

wδ(z − w) + 2T (w)∂wδw(z − w) + ∂wT (w) · δ(z − w).

Therefore

(z − w)4[T (z), T (w)] = 0.

Thus, we obtain that the vertex operator Y (L−2 |0〉 , z) is local with respectto itself. Hence we can apply the reconstruction theorem to give Virc a vertexalgebra structure.

Theorem 3.1.1 The above structures endow Virc with a vertex algebra struc-ture. The defining OPE is

T (z)T (w) ∼ c/2(z − w)4

+T (w)

(z − w)2+∂wT (w)(z − w)

.

The number c is known as the central charge.


3.1.4 Conformal vertex algebras

The Virasoro algebra plays a prominent role in the theory of vertex algebrasand in conformal field theory. Its Lie subalgebra DerO = C[[t]]∂t may beviewed as the Lie algebra of infinitesimal changes of variables on the discD = Spec C[[t]], which is an important symmetry that we will extensively usebelow. Moreover, the full Virasoro algebra “uniformizes” the moduli spacesof pointed curves (see Chapter 17 of [FB]). Often this symmetry is realizedas an “internal symmetry” of a vertex algebra V , meaning that there is anaction of the Virasoro algebra on V that is generated by Fourier coefficientsof a vertex operator Y (ω, z) for some ω ∈ V . This prompts the followingdefinition.

A Z–graded vertex algebra V is called conformal, of central charge c,if we are given a non-zero conformal vector ω ∈ V2 such that the Fouriercoefficients LV

n of the corresponding vertex operator

Y (ω, z) =∑n∈Z

LVn z−n−2 (3.1.5)

satisfy the defining relations of the Virasoro algebra with central charge c, andin addition we have LV

−1 = T , LV0 |Vn

= n Id. Note that we have LVn = ω(n+1).

An obvious example of a conformal vertex algebra is Virc itself, with ω =L−2|0〉. Another example is Vκ(g), with κ 6= κc. In this case, as we have seenabove, ω = S gives it the structure of a conformal vertex algebra with centralcharge cκ.

It is useful to note that a conformal vertex algebra V is automaticallyequipped with a homomorphism φ : Virc → V , which is uniquely determinedby the property that φ(L−2|0〉) = ω.

Here by a vertex algebra homomorphism we understand a linear mapφ : V → V ′, where V and V ′ are vertex algebras (homogeneous of degree 0 ifV and V ′ are Z-graded), such that

(i) φ sends |0〉 to |0〉′;(ii) φ intertwines T and T ′, i.e., φ T = T ′ φ;(iii) φ intertwines Y and Y ′, i.e.,

φ Y (A, z)B = Y ′(φ(A), z)φ(B).

Lemma 3.1.2 A Z+–graded vertex algebra V is conformal, of central chargec, if and only if it contains a non-zero vector ω ∈ V2 such that the Fouriercoefficients LV

n of the corresponding vertex operator

Y (ω, z) =∑n∈Z

LVn z−n−2

satisfy the following conditions: LV−1 = T , LV

0 is the grading operator andLV

2 · ω = c2 |0〉.


Moreover, in that case there is a unique vertex algebra homomorphismVirc → V such that |0〉 7→ |0〉 and L−2vc 7→ ω.

The proof is left to the reader (see [FB], Lemma 3.4.5, where one needs tocorrect the statement by replacing “Z-graded” by “Z+-graded”).

3.1.5 Digression: Why do central extensions keep appearing?

It seems like an interesting coincidence that the Lie algebras associated to thepunctured disc D× = Spec C((t)), such as the formal loop algebra g((t)) andthe Lie algebra DerK of vector fields on D× come with a canonical centralextension. It is therefore natural to ask whether there is a common reasonbehind this phenomenon.

It turns out that the answer is “yes.” This is best understood in terms of acertain “master Lie algebra” that encompasses all of the above examples. ThisLie algebra is gl∞, whose elements are infinite matrices (aij)i,j∈Z with finitelymany non-zero diagonals. The commutator is given by the usual formula[A,B] = AB − BA. This is bigger than the naıve definition of gl∞ as thedirect limit of the Lie algebras gln (whose elements are infinite matrices withfinitely many non-zero entries).

If g is a finite-dimensional Lie algebra, then g⊗C[t, t−1] acts on itself via theadjoint representation. Choosing a basis in g, we obtain a basis in g⊗C[t, t−1]labeled by the integers. Therefore to each element of g ⊗ C[t, t−1] we attachan infinite matrix, and it is clear that this matrix has finitely many non-zero diagonals (but infinitely many non-zero entries!). Thus, we obtain a Liealgebra homomorphism g⊗ C[t, t−1]→ gl∞, which is actually injective.

Likewise, the adjoint action of the Lie algebra Der C[t, t−1] = C[t, t−1]∂t onitself also gives rise to an injective Lie algebra homomorphism Der C[t, t−1]→gl∞.

Now the point is that gl∞ has a one-dimensional universal central extension,and restricting this central extension to the Lie subalgebras g⊗ C[t, t−1] andDer C[t, t−1], we obtain the central extensions corresponding to the affine Kac–Moody and Virasoro algebras, respectively.†

The two-cocycle defining this central extension may be defined as follows.Let us write the matrix A as(

A−+ A++

A−− A+−

).

Then the two-cocycle for the extensions is given by

γ(A,B) = Tr(A++B−− −B++A−−).

† More precisely, of their Laurent polynomial versions, but if we pass to a completion ofegl∞, we obtain in the same way the central extensions of the corresponding completetopological Lie algebras.


It is easy to check that the expression in brackets has finitely many non-zeroentries, and so the trace is well-defined. Since we can regard our Lie algebrasas embedded inside gl∞, we obtain a central extension for each of them byusing the central extension of gl∞. If we work out what these extensions arein our cases, they turn out to be exactly the central extensions we have beenusing. This gives us an explanation of the similarity between the structuresof the Virasoro and Kac–Moody central extensions.

3.2 Lie algebras associated to vertex algebras

Let us summarize where we stand now in terms of reaching our goal of de-scribing the center of the completed universal enveloping algebra Uκ(g).

We have constructed explicitly quadratic elements Sn of Uκ(g), the Segal–Sugawara operators, which were our first candidates for central elements.Moreover, we have computed in Section 3.1 the commutation relations be-tween the Sn’s and gκ and between the Sn’s themselves. In particular, wesaw that these operators commute with gκ when κ = κc.

However, there is an important subtlety which we have not yet discussed.Our computation in Section 3.1 gave the commutation relations betweenFourier coefficients of vertex operators coming from the vertex algebra Vκ(g).By definition, those Fourier coefficients are defined as endomorphisms ofVκ(g), not as elements of Uκ(g). Thus, a priori our computation is onlyvalid in the algebra EndVκ(g), not in Uκ(g). Nevertheless, we will see in thissection that the same result is valid in Uκ(g). In particular, this will implythat the Segal–Sugawara operators are indeed central elements of Uκ(g).

This is not obvious. What is fairly obvious is that there is a natural Liealgebra homomorphism Uκ(g) → EndVκ(g). If this homomorphism were in-jective, then knowing the commutation relations in EndVκ(g) would suffice.But precisely at the critical level κ = κc this homomorphism is not injective.Indeed, the Segal–Sugawara operators Sn, n ≥ −1, annihilate the vacuumvector |0〉 ∈ Vκc

(g), and, because they are central, they are mapped to 0 inEndVκc

(g). Therefore we need to find a way to interpret our computationsof the relations between Fourier coefficients of vertex operators in such a waythat it makes sense inside Uκ(g) rather than EndVκc(g).

In this section we follow closely the material of [FB], Sects. 4.1–4.3.

3.2.1 Lie algebra of Fourier coefficients

Let us define, following [Bo], for each vertex algebra V , a Lie algebra U(V )that is spanned by formal symbols A[n], A ∈ V, n ∈ Z, and whose Lie bracketemulates formula (2.3.8).

3.2 Lie algebras associated to vertex algebras 71

Here is a more precise definition. Let U ′(V ) be the vector space

V ⊗ C[t, t−1]/Im ∂, where ∂ = T ⊗ 1 + 1⊗ ∂t.

We denote by A[n] the projection of A⊗ tn ∈ V ⊗C[t, t−1] onto U ′(V ). ThenU ′(V ) is spanned by these elements modulo the relations

(TA)[n] = −nA[n−1]. (3.2.1)

We define a bilinear bracket on U ′(V ) by the formula

[A[m], B[k]] =∑n≥0

(m

n

)(A(n)B)[m+k−n] (3.2.2)

(compare with formula (2.3.8)). Using the identity

[T,A(n)] = −nA(n−1)

in EndV , which follows from the translation axiom, one checks easily thatthis map is well-defined.

If V is Z–graded, then we introduce a Z–gradation on U ′(V ), by setting,for each homogeneous element A of V , degA[n] = −n+ degA− 1. The map(3.2.2) preserves this gradation.

We also consider a completion U(V ) of U ′(V ) with respect to the naturaltopology on C[t, t−1]:

U(V ) = (V ⊗ C((t)))/ Im ∂.

It is spanned by linear combinations∑

n≥N fnA[n], fn ∈ C, A ∈ V,N ∈ Z,modulo the relations which follow from the identity (3.2.1). The bracket givenby formula (3.2.2) is clearly continuous. Hence it gives rise to a bracket onU(V ).

We have a linear map U(V )→ EndV ,∑n≥N

fnA[n] 7→∑n≥N

fnA(n) = Resz=0 Y (A, z)f(z)dz,

where f(z) =∑

n≥N fnzn ∈ C((z)).

Proposition 3.2.1 The bracket (3.2.2) defines Lie algebra structures onU ′(V ) and on U(V ). Furthermore, the natural maps U ′(V ) → EndV andU(V )→ EndV are Lie algebra homomorphisms.

Proof. Let us define U ′(V )0 ⊂ U(V ) as the quotient

U ′(V )0 = V/ImT = (V ⊗ 1)/ Im ∂,

so it is the subspace of U ′(V ) spanned by the elements A[0] only.The bracket (3.2.2) restricts to a bracket on U ′(V )0:

[A[0], B[0]] = (A(0)B)[0]. (3.2.3)


We first prove that it endows U ′(V )0 with the structure of a Lie algebra underthe defined bracket. We need to show that it is antisymmetric and satisfiesthe Jacobi identity. Recall the identity

Y (A, z)B = ezTY (B,−z)A

from Proposition 2.3.2. Looking at the z−1 coefficient gives

A(0)B = −B(0)A+ T (· · · ).

Taking the [0] part of this and recalling that we mod out by the image of T ,we see that

(A(0)B)[0] = −(B(0)A)[0].

In other words, the bracket is antisymmetric.The Jacobi identity is equivalent to

[C[0], [A[0], B[0]]] = [[C[0], A[0]], B[0]] + [A[0], [C[0], B[0]]].

The right hand side of this gives

((C(0)A)(0)B)[0] + (A(0)(C(0)B))[0].

But from the commutation relations (2.3.8) for the Fourier coefficients ofvertex operators we know that

[C(0), A(0)] = (C(0)A)(0).

This shows the Jacobi identity.We will now derive from this that the bracket (3.2.2) on U ′(V ) satisfies the

axioms of a Lie algebra by constructing a bigger vertex algebra W such thatU ′(V ) is U ′(W )0 (which we already know to be a Lie algebra).

As C[t, t−1] is a commutative, associative, unital algebra with the derivationT = −∂t, we can make it into a commutative vertex algebra, as explained inSection 2.2.2. Let W = V ⊗C[t, t−1] with the vertex algebra structure definedin the obvious way. Then U ′(V ) = U ′(W )0. We now show that formula (3.2.2)in U ′(V ) coincides with formula (3.2.3) in U ′(W )0. The latter reads

[(A⊗ tn)[0], (B ⊗ tm)[0]] = ((A⊗ tn)(0)(B ⊗ tm))[0].

Suppose that f(t) is an element of C[t, t−1]. Then the vertex operator associ-ated to f(t) is

Y (f(t), z) = mult(ezT f(t)) = mult(f(t+ z)),

where necessary expansions are performed in the domain |z| < |t|. In our casethis means that we need the z−1 coefficient of

Y (A, z)⊗ (t+ z)m,


which is ∑n≥0

(m

n

)A(n) ⊗ tm−n.

Putting this together gives

[(A⊗ tm)[0], (B ⊗ tk)[0]] =∑n≥0

(m

n

)(A(n)B)⊗ tm+k−n,

which coincides with formula (3.2.2) we want for U ′(V ).The fact that the map U ′(V ) → EndV is a Lie algebra homomorphism

follows from the comparison of formulas (2.3.8) and (3.2.2). The fact that theLie bracket on U ′(V ) extends to a Lie bracket on U(V ) and the correspondingmap U(V )→ EndV is also a Lie algebra homomorphism follow by continuity.This completes the proof.

Given a vertex algebra homomorphism φ : V → V ′, we define φ to be themap U(V ) → U ′(V ) sending A[n] 7→ φ(A)[n] (and extended by continuity).This is a Lie algebra homomorphism. To see that, we need to check that

φ([A[n], B[m]]) = [φ(A[n]), φ(B[m])].

Expanding the left hand side, using the definition of the bracket, and applyingthe definition of φ to each side reduces it to

φ(A(n)B) = φ(A)(n)φ(B).

But this follows from the fact that φ is a vertex algebra homomorphism.Thus, we obtain that the assignments V 7→ U ′(V ) and V 7→ U(V ) define

functors from the category of vertex algebras to the categories of Lie algebrasand complete topological Lie algebras, respectively.

Finally, we note that if V is a Z-graded vertex algebra, then the Lie algebraU ′(V ) is also Z-graded with respect to the degree assignment degA[n] =−n+ degA− 1.

3.2.2 From vertex operators to the enveloping algebra

Let us now look more closely at the Lie algebra U(Vκ(g)). By Theorem 2.2.2,the vertex operators corresponding to elements of Vκ(g) are finite linear com-binations of the formal power series

:∂n1z Ja1(z) · · · ∂nm

z Jam(z): .

This means that we may think of elements of U(Vκ(g)) as finite linear com-binations of Fourier coefficients of these series. Each of these coefficients is,in general, an infinite series in terms of the basis elements Ja

n of g. However,one checks by induction that each of these series becomes finite modulo theleft ideal Uκ(g) ·g⊗ tNC[[t]], for any N ≥ 0. Therefore each of them gives rise


to a well-defined element of Uκ(g). We have already seen that in the case ofthe Segal–Sugawara elements (2.1.9): they are bona fide elements of Uκ(g).

Thus, we have a linear map V ⊗ C((t)) → Uκ(g), and it is clear that itvanishes on the image of ∂. Therefore we obtain a linear map U(Vκ(g)) →Uκ(g). It is not difficult to show that it is actually injective, but we will notuse this fact here. The key result which will allow us to transform our previouscomputations from EndV to Uκ(g) is the following:

Proposition 3.2.1 The map U(Vκ(g)) → Uκ(g) is a Lie algebra homomor-phism.

The proof involves some rather tedious calculations, so we will omit it,referring the reader instead to [FB], Proposition 4.2.2.

Now we see that all of our computations in Section 3.1 are in fact valid inUκ(g).

Corollary 3.2.2 For κ = κc the Segal–Sugawara operators Sn are central inUκ(g). For κ 6= κc they generate the Virasoro algebra inside Uκ(g).

3.2.3 Enveloping algebra associated to a vertex algebra

Here we construct an analogue of the topological associative algebra Uκ(g) foran arbitrary Z-graded vertex algebra V .

Note that the image of U(Vκ(g)) in Uκ(g) is not closed under multiplica-tion. For instance, U(Vκ(g)) contains g as a Lie subalgebra. It is spanned byAn = (A−1|0〉)[n], A ∈ g, n ∈ Z, and (|0〉)[−1]. But U(Vκ(g)) does not containproducts of elements of g, such as AnBk. In fact, all quadratic elements inU(Vκ(g)) are infinite series.

However, we can obtain Uκ(g) as a completion of the universal envelopingalgebra of U(Vκ(g)) modulo certain natural relations. An analogous con-struction may in fact be performed for any Z-graded vertex algebra V in thefollowing way. Denote by U(U(V )) the universal enveloping algebra of theLie algebra U(V ). Define its completion

U(U(V )) = lim←−

U(U(V ))/IN ,

where IN is the left ideal generated by A[n], A ∈ Vm, n ≥ N +m.† Let U(V )be the quotient of U(U(V )) by the two-sided ideal generated by the Fouriercoefficients of the series

Y [A(−1)B, z]− :Y [A, z]Y [B, z]:, A,B ∈ V,

† Here we correct an error in [FB], Section 4.3.1, where this condition was written asn > N ; the point is that deg A[n] = −n + m− 1 (see the end of Section 3.2.1).


where we set

Y [A, z] =∑n∈Z

A[n]z−n−1,

and the normal ordering is defined in the same way as for the vertex operatorsY (A, z).

Clearly, U(V ) is a complete topological associative algebra (with the topol-ogy for which a basis of open neighborhoods of 0 is given by the completions ofthe left ideals IN , N > 0). The assignment V 7→ U(V ) gives rise to a functorfrom the category of vertex algebras to the category of complete topologicalassociative algebras.

The associative algebra U(V ) may be viewed as a generalization of thecompleted enveloping algebra Uκ(g), because, as the following lemma shows,for V = Vκ(g) we have U(Vκ(g)) ' Uκ(g).

Lemma 3.2.2 There is a natural isomorphism U(Vκ(g)) ' Uκ(g) for allκ ∈ C.

Proof. We need to define mutually inverse algebra homomorphisms be-tween U(Vκ(g)) and Uκ(g). The Lie algebra homomorphism U(Vκ(g)) →Uκ(g) of Proposition 3.2.1 gives rise to a homomorphism of the universal en-veloping algebra of the Lie algebra U(Vκ(g)) to Uκ(g). It is clear that underthis homomorphism any element of the ideal IN is mapped to the left idealof U(Vκ(g)) generated by An, A ∈ g, n > M , for sufficiently large M . There-fore this homomorphism extends to a homomorphism from the completionof the universal enveloping algebra of U(Vκ(g)) to Uκ(g). But according tothe definitions, this map sends the series Y [A(−1)B, z] precisely to the se-ries :Y [A, z]Y [B, z]: for all A,B ∈ Vκ(g). Hence we obtain a homomorphismU(Vκ(g))→ Uκ(g).

To construct the inverse homomorphism, recall that g is naturally a Liesubalgebra of U(Vκ(g)). Hence we have a homomorphism from Uκ(g) to theuniversal enveloping algebra of the Lie algebra U(Vκ(g)). This homomorphismextends by continuity to a homomorphism from Uκ(g) to the completion of theuniversal enveloping algebra of U(Vκ(g)), and hence to U(Vκ(g)). Now observethat the resulting map Uκ(g) → U(Vκ(g)) is surjective, because each seriesY [A, z] is a linear combination of normally ordered products of the generatingseries Ja(z) =

∑n∈Z J

na z−n−1 of the elements Ja

n of g. Furthermore, it isclear from the construction that the composition Uκ(g)→ U(Vκ(g))→ Uκ(g)is the identity. This implies that the map Uκ(g)→ U(Vκ(g)) is also injective.Therefore the two maps indeed define mutually inverse isomorphisms.


3.3 The center of a vertex algebra

According to Corollary 3.2.2, the Segal–Sugawara elements are indeed centralelements of the completed enveloping algebra Uκ(g) for a special level κ = κc.Now we would like to see if there are any other central elements, and, if so,how to construct them.

3.3.1 Definition of the center

In order to do this we should think about why the elements we constructedwere central in the first place.

Lemma 3.3.1 The elements A[n], A ∈ Vκ(g), n ∈ Z, of Uκ(g) are central if Ais annihilated by all operators Ja

n with n > 0.

Proof. This follows immediately from the commutation relations formula(3.2.2).

The elements Jan , n ≥ 0, span the Lie algebra g[[t]]. This leads us to de-

fine the center z(Vκ(g)) of the vertex algebra Vκ(g) as its subspace of g[[t]]-invariants:

z(Vκ(g)) = Vκ(g)g[[t]]. (3.3.1)

In order to generalize this definition to other vertex algebras, we note thatany element B ∈ z(Vκ(g)) satisfies A(n)B = 0 for all A ∈ Vκ(g) and n ≥ 0.This follows from formula (2.2.8) and the definition of normal ordering byinduction on m.

Thus, we define the center of an arbitrary vertex algebra V to be

Z(V ) = B ∈ V |A(n)B = 0 for all A ∈ V, n ≥ 0.

By the OPE formula (2.3.7), an equivalent definition is

Z(V ) = B ∈ V | [Y (A, z), Y (B,w)] = 0 for all A ∈ V .

Formula (3.2.2) implies that if B ∈ Z(V ), then B[n] is a central element inU(V ) and in U(V ) for all n ∈ Z.

The space Z(V ) is clearly non-zero, because the vacuum vector |0〉 is con-tained in Z(V ).

Lemma 3.3.2 The center Z(V ) of a vertex algebra V is a commutative vertexalgebra.

Proof. We already know that |0〉 is in the center. A simple calculationshows that T maps Z(V ) into Z(V ):

A(n)(TB) = T (A(n)B)− [T,A(n)]B = nA(n−1)B = 0.

3.3 The center of a vertex algebra 77

Let B,C ∈ Z(V ) be central. Then

A(n)B(m)C = B(m)A(n)C + [A(n), B(m)]C = 0.

Therefore Z(V ) is closed under the state-field correspondence. The commu-tativity is obvious.

3.3.2 The center of the affine Kac–Moody vertex algebra

So our plan is now to study the center of the vertex algebra Vκ(g) and thento show that we obtain all central elements for Uκ(g) from the elements inZ(Vκ(g)) by taking their Fourier coefficients (viewed as elements of Uκ(g)).

Since Vκ(g) is much smaller than Uκ(g), this looks like a good strategy. Westart with the case when κ 6= κc.

Proposition 3.3.3 If κ 6= κc, then the center of Vκ(g) is trivial, i.e.,

Z(Vκ(g)) = C · |0〉 .

Proof. Suppose that A is a homogeneous element in the center Z(Vκ(g)).Let S be the vector introduced above generating the Virasoro field

Y (S, z) =∑n∈Z

Lnz−n−2.

Recall that we need κ 6= κc for this element to be well-defined as we needed todivide by this factor. Now, if B ∈ Z(Vκ(g)), then A(n)B = 0 for all A ∈ Vκ(g)and n ≥ 0, as we saw above. In particular, taking A = S and n = 1 we findthat L0B = 0 (recall that L0 = S(1)). But L0 is the grading operator, so wesee that B must have degree 0, hence it must be a multiple of the vacuumvector |0〉.

Note that a priori it does not follow that the center of Uκ(g) for κ 6= κc istrivial. It only follows that we cannot obtain non-trivial central elements ofUκ(g) from the vertex algebra Vκ(g) by the above construction. However, wewill prove later, in Proposition 4.3.9, that the center of Uκ(g) is in fact trivial.

From now on we will focus our attention on the critical level κ = κc. Wewill see that the center of Uκc

(g) may be generated from the center of thevertex algebra Vκc

(g). From now on, in order to simplify the notation we willdenote the latter by z(g), so that

z(g) = Z(Vκc(g)).

We now discuss various interpretations of z(g).The commutative vertex algebra structure on z(g) gives rise to an ordinary

commutative algebra structure on it. Note that it is Z+-graded. It is naturalto ask what the meaning of this algebra structure is from the point of view ofVκc(g).


According to (3.3.1), we have

z(g) = Vκc(g)g[[t]]. (3.3.2)

This enables us to identify z(g) with the algebra of gκc -endomorphisms ofVκc

(g). Indeed, a g[[t]]-invariant vector A ∈ Vκc(g) gives rise to a non-trivial

endomorphism of V , which maps the highest weight vector vκcto A. On the

other hand, given an endomorphism e of Vκc(g), we obtain a g[[t]]-invariantvector e(|0〉). It is clear that the two maps are inverse to each other, and sowe obtain an isomorphism

Vκc(g)g[[t]] ' EndbgκcVκc(g).

Now let A1 and A2 be two g[[t]]-invariant vectors in Vκc(g). Let e1 and e2

be the corresponding endomorphisms of Vκc(g). Then the image of |0〉 underthe composition e1e2 equals

e1(A2) = e1 (A2)(−1)|0〉 = (A2)(−1) e1(|0〉) = (A2)(−1)A1.

The last expression is nothing but the product A2A1 coming from the com-mutative algebra structure on z(g). Therefore we find that as an algebra, z(g)is isomorphic to the algebra Endbgκc

Vκc(g) with the opposite multiplication.But since z(g) is commutative, we obtain an isomorphism of algebras

z(g) ' EndbgκcVκc(g). (3.3.3)

In particular, we find that the algebra EndbgκcVκc(g) is commutative, which

is a priori not obvious. (The same argument also works for other levels κ,but in those cases we find that z(Vκ(g)) ' Endbgκ

Vκ(g) = C, according toProposition 3.3.3.)

Finally, we remark that Vκc(g) is isomorphic to the universal enveloping

algebra U(g⊗t−1C[t−1]) (note however that this isomorphism is not canonical:it depends on the choice of the coordinate t). Under this isomorphism, weobtain an injective map

z(g) → U(g⊗ t−1C[t−1]).

It is clear from the above calculation that this is a homomorphism of algebras.Thus, z(g) may be viewed as a commutative subalgebra of U(g⊗ t−1C[t−1]).

3.3.3 The associated graded algebra

Our goal is to describe the center z(g) of Vκc(g). It is instructive to describefirst its graded analogue. Recall that in Section 2.2.5 we defined a filtrationon Vκ(g) that is preserved by the action of gκ and such that the associatedgraded space is

grVκ(g) = Sym(g((t))/g[[t]]).

3.3 The center of a vertex algebra 79

The topological dual space to g((t))/g[[t]] is naturally identified with the spaceg∗[[t]]dt with respect to the pairing

〈φ(t)dt,A(t)〉 = Rest=0〈φ(t), A(t)〉dt.

This pairing is canonical in the sense that it does not depend on the choice ofcoordinate t.

However, to simplify our formulas below, we will use a coordinate t toidentify g∗[[t]]dt with g∗[[t]]. This is the inverse limit of finite-dimensionalvector spaces

g∗[[t]] = lim←−

g∗ ⊗ C[[t]]/(tN ),

and so the algebra of regular functions on it is the direct limit of free polyno-mial algebras

Fun g∗[[t]] = lim−→

Fun(g∗ ⊗ C[[t]]/(tN )).

Since g∗[[t]] is isomorphic to the topological dual vector space to g((t))/g[[t]],we have a natural isomorphism

Sym(g((t))/g[[t]]) ' Fun g∗[[t]],

and so we have

grVκ(g) ' Fun g∗[[t]].

Now the filtration on Vκc(g) induces a filtration on z(g). Let

gr z(g) ⊂ Fun g∗[[t]]

be the associated graded space. The commutative algebra structure on z(g)gives rise to such a structure on gr z(g). Note also that the Lie algebra g[[t]]acts on itself via the adjoint representation, and this induces its action onFun g∗[[t]].

Lemma 3.3.1 We have an injective homomorphism of commutative algebras

gr z(g) → (Fung∗[[t]])g[[t]].

Proof. The fact that the embedding gr z(g) → Fun g∗[[t]] is a homomor-phism of algebras follows from the isomorphism (3.3.3).

Suppose now that A ∈ Vκc(g)6i, and denote the projection of A ontogri(Vκc(g)) = Vκc(g)6i/Vκc(g)6(i−1) by Symbi(A). It is easy to see that

Symbi(x ·A) = x · Symbi(A) for x ∈ g[[t]].

If A ∈ z(g), then x ·A = 0 and so x · Symbi(A) = 0.

Thus, the symbols of elements of z(g) are realized as g[[t]]-invariant functions


on g∗[[t]]. We now begin to study the algebra of these functions, which wedenote by

Inv g∗[[t]] = (Fun g∗[[t]])g[[t]].

Note that the action of the Lie algebra g[[t]] on Fun g∗[[t]] comes from thecoadjoint action of g on g∗. This action may be exponentiated to the adjointaction of the group G[[t]], where G is the connected simply-connected Liegroup with Lie algebra g. Therefore G[[t]] acts on Fun g∗[[t]]. Since G[[t]] isconnected, it follows that

Inv g∗[[t]] = (Fun g∗[[t]])G[[t]].

If we can work out the size of this algebra of invariant functions, then wewill be able to place an upper bound on the size of z(g) and this will turn outto be small enough to allow us to conclude that the elements we subsequentlyconstruct give us the entire center of Vκc(g) (and ultimately of Uκc(g)).

3.3.4 Symbols of central elements

It is instructive to look at what happens in the case of a finite-dimensionalsimple Lie algebra g. In this case the counterpart of z(g) is the center Z(g)of the universal enveloping algebra U(g) of g. Let grU(g) be the associatedgraded algebra with respect to the Poincare–Birkhoff–Witt filtration. Then

grU(g) = Sym g = Fun g∗.

Let grZ(g) be the associated graded algebra of Z(g) with respect to theinduced filtration. In the same way as in Lemma 3.3.1 we obtain an injectivehomomorphism

grZ(g) → Inv g∗ = (Fun g∗)g = (Fun g∗)G.

It is actually an isomorphism, because U(g) and Sym g are isomorphic asg-modules (with respect to the adjoint action). This follows from the factthat both are direct limits of finite-dimensional representations, and there areno non-trivial extensions between such representations.

Recall that by Theorem 2.1.1 we have

Z(g) = C[Pi]i=1,...,`.

Let P i be the symbol of Pi in Fun g∗. Then

Inv g∗ = C[P i]i=1,...,`.

According to Theorem 2.1.1, degP i = di + 1.We will now use the elements P i to create a large number of elements

in Inv g∗[[t]]. We will use the generators Ja

n, n < 0, of Sym(g((t))/g[[t]]) =

3.4 Jet schemes 81

Fun g∗[[t]], which are the symbols of Jan |0〉 ∈ Vκc(g). These are linear functions

on g∗[[t]] defined by the formula

Ja

n(φ(t)) = Rest=0〈φ(t), Ja〉tndt. (3.3.4)

We will also write

Ja(z) =

∑n<0

Ja

nz−n−1.

Suppose that we write P i as a polynomial in the linear elements, P i(Ja).

Define a set of elements P i,n ∈ Fun g∗[[t]] by the formula

P i(Ja(z)) =

∑n<0

P i,nz−n−1.

Note that each of the elements P i,n is a finite polynomial in the Ja

n’s.

Lemma 3.3.4 The polynomials P i,n are in Inv g∗[[t]].

The proof is straightforward and is left to the reader.Thus, we have a homomorphism

C[P i,n]i=1,...,`;n<0 −→ Inv g∗[[t]].

We wish to show that this is actually an isomorphism, but in order to do thiswe need to develop some more technology.

3.4 Jet schemes

Let X be a complex algebraic variety. A jet scheme of X is an algebraicobject that represent the “space of formal paths on X”, i.e., morphisms fromthe disc D = Spec C[[t]] to X. We wish to treat it as a scheme, i.e., somethingglued from affine algebraic varieties defined by polynomial equations in anaffine space. In this section we give a precise definition of these objects. Weare interested in these objects because of the simple fact that g∗[[t]] is the jetscheme of g∗, considered as an affine algebraic variety. The technique of jetschemes will allow us to describe the algebra of invariant functions on g∗[[t]].

3.4.1 Generalities on schemes

In algebraic geometry it is often convenient to describe the functor of pointsof an algebraic variety (or a scheme). For example, an affine algebraic varietyX over a field k is completely determined by the ring of regular functions onit. Let us denote it by FunX, so we can write X = Spec(FunX). Now, givena k-algebra R, we define the set of R-points of X, denoted by X(R), as theset of morphisms SpecR→ X. This is the same as the set of homomorphismsFunX → R of k-algebras.


If R = k, we recover the usual notion of k-points of X, because each ho-momorphism FunX → k corresponds to taking the value of a function at apoint x ∈ X(k). However, the set of k-points by itself does not capture theentire structure of X in general. For example, if k is not algebraically closed,the set X(k) may well be empty, but this does not mean that X is empty.Consider the case of X = Spec R[x]/(x2 +1). Then X(R) is empty, but X(C)consists of one element. Therefore we see that we may well lose informationabout X if we look at its k-points. If X is an affine algebraic variety, then it isthe algebra FunX which captures the structure of X, not the set X(k). For ageneral algebraic variety X, it is the structure of a scheme on X, which essen-tially amounts to a covering of X by affine algebraic varieties, each describedby its algebra of regular functions.

Now, fixing X, but letting the k-algebra R vary, we obtain a functor fromthe category of commutative k-algebras to the category of sets which satisfiessome natural conditions (it is a “sheaf” on the category of k-algebras, in theappropriate sense). These conditions ensure that it extends uniquely to afunctor from the category of schemes over k (as those are the objects “glued”from the affine algebraic varieties) to the category of sets. This is what isreferred to as the “functor of points.”

One defines in a similar way the functor of points of an arbitrary algebraicvariety X (not necessarily affine) over k. An important result is that thisfunctor determines X uniquely (up to isomorphism). Such functors from thecategory of k-algebras (or from the category of schemes over k) to the categoryof sets are called representable, and in that case we say that the variety X

represents the corresponding functor. Thus, if we want to define a “would be”algebraic variety X, we do not lose any information if we start by defining its“would be” functor of points. A priori, it may be unclear whether the functoris a functor of points (i.e., that it may be realized by the above construction).So the question we need to answer is whether this functor is representable. Ifit is, then the object that represents it is the sought-after algebraic variety.

3.4.2 Definition of jet schemes

Let us see now how this works in the case of jet schemes. In this case the“would be” functor of points is easy to define. Indeed, let X is an algebraicvariety, which for definiteness we will assume to be defined over C. Then anR-point of X is a morphism SpecR→ X, or equivalently a homomorphism ofalgebras FunX → R. An R-point of the “would be” jet scheme JX should beviewed as a morphism from the disc over Spec(R), which is Spec(R[[t]]), to X.Thus, we define the functor of points of JX as the functor sending a C-algebraR to the set of morphisms Spec(R[[t]]) → X, and homomorphisms R → R′

to obvious maps between these sets. It is not difficult to see that this functoris a “sheaf,” and so in particular extends in a unique way to the category of

3.4 Jet schemes 83

all schemes over C. In what follows we will just look at this functor on thecategory of C-algebras.

If X is an affine variety, then a morphism SpecR[[t]] → X is the same asa homomorphism FunX → R[[t]], so we obtain a very concrete realization ofthe “would be” functor of points of the set scheme JX.

For example, suppose that X is the affine space AN . Choosing coordinatefunctions x1, . . . , xN on AN , we obtain that Fun AN = C[x1, . . . , xN ]. In otherwords, AN = Spec C[x1, . . . , xN ]. Thus, C-points of AN are homomorphismsφ : C[x1, . . . , xN ] → C. Those are given by n-tuples of complex numbers(a1, . . . , aN ), such that φ(xi) = ai, as expected. Likewise, its R-points aren-tuples of elements of R.

By definition, the set of R-points of the jet scheme JAN should now begiven by homomorphisms

C[x1, . . . , xN ]→ R[[t]].

Such a homomorphism is uniquely determined by the images of the generatorsxi, which are now formal power series

ai(t) =∑n<0

ai,nt−n−1, ai,n ∈ R, i = 1, . . . , N (3.4.1)

(the indices are chosen so as to agree with the convention of our previousformulas). Note that there are no conditions of convergence on the formalpower series. In other words, it is really “formal” paths that we are describing.

We can equivalently think of the collection of N formal power series as acollection of all of their Fourier coefficients. Hence the set of R-points of JAN

is

ai,n ∈ R | i = 1, . . . , N ;n < 0.

The corresponding functor is representable by

JAN def= Spec C[xi,n]i=1,...,N ;n<0.

Thus we obtain the definition of the jet scheme JAN of AN . As expected,JAN is an infinite-dimensional affine space (inverse limit of finite-dimensionalones).

Suppose now that we have a finite-dimensional affine algebraic variety X.What does its jet scheme look like? We may write

X = Spec (C[x1, . . . , xN ]/ 〈F1, . . . , FM 〉) ,

where the Fi’s are polynomials in the variables x1, . . . , xN . This means thatthe complex points of X are n-tuples of complex numbers (a1, . . . , aN ) satis-fying the equations

Fi(a1, . . . , aN ) = 0 for all i.


By definition, the set of R-points of the jet scheme JX of X is the set ofhomomorphisms

C[x1, . . . , xN ]/ 〈F1, . . . , FM 〉 → R[[t]].

Such a homomorphism is again uniquely determined by the images of the xi’swhich are formal power series ai(t) as in formula (3.4.1), but now they haveto satisfy the equations

Fi(a1(t), . . . , aN (t)) ≡ 0, i = 1, . . . , N. (3.4.2)

We obtain the equations on the Fourier coefficients ai,n by reading off thecoefficients of tm.

A compact way to organize these equations is as follows. Define a derivationT of the algebra C[xj,n] by the formula

T (xj,n) = −nxj,n−1.

Let us replace xj by xj,−1 in the defining polynomials Fi and call these Fi

again. Then the equations that come from (3.4.2) are precisely

TmFi = 0, i = 1, . . . , N ;m ≥ 0.

Therefore we find that this functor of points is representable by the scheme

JX = Spec(C[xj,n]j=1,...,N ;n<0/ 〈TmFi〉i=1,...,M ;m>0

).

This is the jet scheme of X.In particular, we see that the algebra of functions on JX carries a derivation

T , inherited from C[xj,n]j=1,...,N ;n<0. Therefore it is a commutative vertexalgebra.

Now we have the definition of the jet scheme of an arbitrary affine scheme.What about more general schemes? An arbitrary scheme has a covering byaffine schemes. To define the jet scheme of X we simply need to work outhow to glue together the jet schemes for the affine pieces. For example P1

can be decomposed into two affine schemes A1 ∪ A1, where these are thecomplements of∞ and 0 respectively. Therefore they are given by Spec(C[x])and Spec(C[y]) with identification given by x↔ y−1. We already know how toget the jet schemes of these affine components. The obvious way to generalizethe gluing conditions is to substitute formal power series into them, i.e.,

x(t)←→ y(t)−1.

In order to write y(t)−1 as a formal power series in t it is only necessary toinvert y−1, which is invertible on the overlap of the two affine lines in P1.Therefore this formula is well-defined on the overlap, and it glues togetherthe jet schemes of the two affine lines into the jet scheme of P1.

A similar statement is true in general and makes the gluing possible (theonly difficulties come from the inverted variables, but these are compatible as

3.4 Jet schemes 85

they were compatible for the original scheme). Therefore we have constructedthe scheme JX associated to any scheme X (of finite type).

Let

Dn = Spec(C[[t]]/(tn+1))

be the nth order neighborhood of a point. We can construct nth order jetschemes JnX which represent the maps from Dn to X. The natural mapsDn−1 → Dn give rise to maps JnX → Jn−1X. We may then represent the(infinite order) jet scheme JX as the inverse limit of the finite jet schemes:

JX = lim←−

JnX.

This gives a realization of the jet scheme as an inverse limit of schemes offinite type.

The following lemma follows immediately from the criterion of smoothness:a morphism φ : X → Y of schemes of finite type over C is smooth if andonly if for any C-algebra R and its nilpotent enlargement Rnilp (i.e., a C-algebra whose quotient by the ideal generated by nilpotent elements is R)and a morphism p : Spec(R)→ Y , which lifts to X via φ, any extension of pto a morphism Spec(Rnilp)→ Y also lifts to X.

We will say that a morphism of C-schemes X → Y is surjective if thecorresponding map of the sets of C-points X(C)→ Y (C) is surjective.

Lemma 3.4.1 If a morphism φ : X → Y is smooth and surjective, then thecorresponding morphisms of jet schemes Jnφ : JnX → JnY and Jφ : JX →JY are (formally) smooth and surjective.

3.4.3 Description of invariant functions on Jg

We now use the notion of jet schemes to prove the following theorem aboutthe invariant functions on g∗[[t]], which is due to A. Beilinson and V. Drinfeld[BD1] (see also Proposition A.1 of [EF]). Recall that g is a simple Lie algebraof rank `.

Theorem 3.4.2 The algebra Inv g∗[[t]] is equal to C[P i,n]i=1,...,`;n<0.

Proof. Recall that Inv g∗[[t]] is the algebra of G[[t]]-invariant functions ong∗[[t]]. In terms of jet schemes, we can write g∗[[t]] = Jg∗, G[[t]] = JG, andso the space Inv g∗[[t]] = (FunJg∗)JG should be thought of as the algebra offunctions on the quotient Jg∗/JG. Unfortunately, this is not a variety, as thestructure of the orbits of JG in Jg∗ is rather complicated.

More precisely, define P to be Spec Inv g∗ = SpecC[P i]i=1,...,`. Thus, Pis the affine space with coordinates P i, i = 1, . . . , `. The inclusion Inv g∗ →Fun g∗ gives rise to a surjective map p : g∗ → P. The group G acts along thefibers, but the fibers do not always consist of single G-orbits.


For example, the inverse image of 0 is the nilpotent cone. Let us identifyg with g∗ by means of a non-degenerate invariant inner product. Then inthe case of g = sln the nilpotent cone consists of all nilpotent n× n matricesin sln. The orbits of nilpotent matrices are parameterized by the Jordanforms, or, equivalently, partitions of n. There is one dense orbit consistingof matrices with one Jordan block and several smaller orbits correspondingto other Jordan block forms. It is these smaller orbits that are causing theproblems. In particular, the morphism g∗ → P is not smooth. So we throwthese orbits away by defining the open dense subset of regular elements

g∗reg = x ∈ g∗ | dim gx = `

where gx is the centralizer of x in g.†The map obtained by restriction

preg : g∗reg −→ P

is already smooth. Furthermore, it is surjective and each fiber is a singleG-orbit [Ko].

Applying the functor of jet schemes, we obtain a morphism

Jpreg : Jg∗reg −→ JP,

which is (formally) smooth and surjective, by Lemma 3.4.1. Here

JP = Spec C[P i,n]i=1,...,`;n<0

is the infinite affine space with coordinates P i,n, i = 1, . . . , `;n < 0.We want to show that the fibers of Jpreg also consist of single JG-orbits.

To prove that, consider the map

G× g∗reg −→ g∗reg ×P

g∗reg.

This is again a smooth morphism. But note now that its surjectivity is equiv-alent to the fact that the fibers of g∗reg −→ P consist of single orbits. Takingjet schemes, we obtain a morphism

JG× Jg∗reg −→ Jg∗reg ×JP

Jg∗reg.

It is (formally) smooth and surjective by Lemma 3.4.1. This shows that eachfiber of Jpreg is a single JG-orbit.

This implies that the algebra of JG-invariant functions on Jg∗reg is equal toFun(JP) = C[P i,n]i=1,...,`;n<0.

To be precise, we prove first the corresponding statement for finite jetschemes. For each N ≥ 0 we have a morphism JNg∗reg → JNP, equippedwith a fiberwise action of JNG. We obtain in the same way as above thateach fiber of this morphism is a single JNG-orbit. Proposition 0.2 of [Mu]

† This is the minimal possible dimension the centralizer can have.

3.5 The center in the case of sl2 87

then implies that the algebra of JNG-invariant functions on JNg∗reg is equalto Fun(JNP) = C[P i,n]i=1,...,`;−N−1≤n<0.

Now, any JG-invariant function on Jg∗reg comes by pull-back from a JNG-invariant function on JNg∗reg. Hence we obtain that the algebra of JG-invariant functions on Jg∗reg is equal to Fun(JP) = C[P i,n]i=1,...,`;n<0.

But regular elements are open and dense in g∗, and hence JNg∗reg is openand dense in JNg∗, for both finite and infinite N . Therefore all functions onJNg∗ are uniquely determined by their restrictions to JNg∗reg, for both finiteand infinite N . Therefore the algebra of JNG-invariant functions on JNg∗ isequal to Fun(JNP) = C[P i,n]i=1,...,`;−N−1≤n<0.

3.5 The center in the case of sl2

Thus, we have now been able to describe the graded analogue gr z(g) of thecenter z(g).

It is now time to describe the center z(g) itself. We start with the simplestcase, when g = sl2.

3.5.1 First description

We know from Lemma 3.3.1 that there is an injection from the associatedgraded gr z(g) of the center z(g) of Vκc(g) into the algebra Inv g∗[[t]] of invariantfunctions on g∗[[t]]. We have just shown that the algebra Inv g∗[[t]] is equalto the polynomial ring C[P i,n]i=1,...,`;n<0. In the case of g = sl2(C) this isalready sufficient to determine the structure of the center z(sl2) of the vertexalgebra Vκc

(sl2). Namely, we prove the following:

Theorem 3.5.1 The center z(sl2) of the vertex algebra Vκc(sl2) is equal toC[Sn]n≤−2 |0〉. Here, the Sn’s are the Segal–Sugawara operators (note thatthe operators Sn, n ≥ −1, annihilate the vacuum vector |0〉).

Proof. We already know that the Segal–Sugawara operators Sn, n ∈ Z, arecentral elements of the completed enveloping algebra of critical level. There-fore we have a map

C[Sn]n≤−2 |0〉 → Vκc(sl2)sl2[[t]] = z(sl2). (3.5.1)

We need to prove that this map is an equality, i.e., the operators Sn, n ≤ −2,are algebraically independent and there are no other elements in z(sl2). Forthat it is sufficient to show that the map of associated graded algebras is anisomorphism.

Recall that

Sn =12

∑a

∑j+k=n

:Jaj Ja,k: . (3.5.2)


For n ≤ −2 some of these terms will have both j and k negative and the restwill have at least one of j and k positive; the former will act

V (g)6i −→ V (g)6(i+2)

(because they are creation operators) and the rest act

V (g)6i −→ V (g)6(i+1)

(because they have at least one annihilation operator). Hence the latter termswill be removed when looking at the symbol, and we find that the symbol ofthe monomial Sn1 . . . Snm with ni ≤ −2 is equal to P 1,n1+1 . . . P 1,nm+1. Here

P 1(z) =∑n<0

P 1,nz−n−1 =

12

∑a

Ja(z)Ja(z)

is the first (and the only, for g = sl2) of the generating series P i(z) consideredabove.

In particular, we see that the map (3.5.1) is compatible with the naturalfiltrations, and the corresponding map of associated graded spaces factors asfollows

gr C[Sn]n≤−2 |0〉 ' C[Pn]n≤−2 → (grVκc(sl2))sl2[[t]] ' Inv(sl2)∗[[t]],

where the first map sends Sn 7→ P 1,n. We obtain from Theorem 3.4.2 thatthe second map is an isomorphism. Therefore we have

gr C[Sn]n≤−2 |0〉 ' Inv(sl2)∗[[t]].

But we know that

gr C[Sn]n≤−2 |0〉 ⊂ gr z(sl2) ⊂ Inv (sl2)∗[[t]]

(see Lemma 3.3.1). Therefore both of the above inclusions are equalities, andso gr C[Sn]n≤−2 |0〉 = gr z(sl2). Since we know that C[Sn]n≤−2 |0〉 ⊂ z(sl2),this implies that C[Sn]n≤−2 |0〉 = z(sl2).

A similar result may be obtained for the center of the completed universalenveloping algebra Uκc

(sl2): the center is topologically generated by Sn, n ∈ Z(so this time all of the Sn’s are used). The argument is essentially the same,but we will postpone this proof till Section 4.3.2.

We will see below that both of these results remain true in the case of amore general Lie algebra g. In other words, we will show that z(g) is “aslarge as possible,” that is, gr z(g) = Inv g∗[[t]]. This is equivalent to theexistence of central elements Si ∈ z(g) ⊂ Vκc(g), whose symbols are equal toP i,−1 ∈ Inv g∗[[t]]. It will then follow in the same way as for g = sl2 that

z(g) = C[Si,[n]]i=1,...,`;n<0,


where the Si,[n]’s are the Fourier coefficients of the formal power series Y [Si, z].Moreover, it will follow that the center of Uκc

(g) is topologically generated bySi,[n], i = 1, . . . , `;n ∈ Z.

We have already constructed the first of these elements; namely, the quadra-tic element 1

2Ja−1Ja,−1|0〉 ∈ z(g) whose symbol is 1

2Ja

−1Ja,−1, corresponding tothe quadratic Casimir element P 1. Unfortunately, the formulas for the centralelements corresponding to the higher order Casimir elements P i,−1, i > 1, areunknown in general (the leading order terms are obvious, but it is finding thelower order terms that causes lots of problems). So we cannot prove this resultdirectly, as in the case of g = sl2. We will have to use a different argumentinstead.

3.5.2 Coordinate-independence

Our current approach suffers from another kind of deficiency: everything wehave done so far has been coordinate-dependent. We started with a Lie algebrag and constructed the Lie algebra g as the central extension of g ⊗ C((t)).However, there are many local fields which are isomorphic to the field C((t)),but not naturally.

For example, let X be a smooth algebraic curve over C, and x any pointon X. The choice of x gives rise to a valuation | · |x on the field of rationalfunctions C(X). The valuation is defined to be the order of vanishing of f atx and is coordinate-independent. The completion of the algebra of functionsunder this valuation is denoted by Kx. Let us pick a local coordinate at x,that is, a rational function vanishing to order 1 at x (more generally, we maychoose a “formal coordinate,” which is an arbitrary element of Kx vanishingat x to order 1). If we denote this function by t, then we obtain a mapC((t)) → Kx which is an isomorphism. Thus, we identify Kx with C((t)) foreach choice of a coordinate at x. But usually there is no preferred choice, andtherefore there is no canonical isomorphism Kx ' C((t)).

However, all of the objects that we have discussed so far may be attachedcanonically to Kx without any reference to a particular coordinate. Indeed,we define the affine Kac–Moody algebra gκ,x attached to Kx as the centralextension

0 −→ C1 −→ gκ,x −→ g⊗Kx −→ 0

given by the two-cocycle

c(A⊗ f,B ⊗ g) = −κ(A,B) Resx(f dg).

This formula is coordinate-independent as f dg is a one-form, and so its residueis canonically defined. We then define a coordinate-independent vacuum gκ,x-module associated to the point x as

Vκ(g)x = Indbgκ,x

g⊗Ox⊕C1C, (3.5.3)


where Ox is the ring of integers of Kx (comprising those elements of Kx whichhave no pole at x).

Now our task is to describe the algebra

z(g)x = EndbgxVκc(g)x = (Vκc(g)x)g⊗Ox , (3.5.4)

and we should try to do this in a coordinate-independent way. Unfortunately,everything we have done so far has been pegged to the field C((t)), and so tointerpret our results for an arbitrary local field Kx we need to pick a coordinatet at x. The problem is that we do not know yet what will happen if we chooseanother coordinate t′. Therefore at the moment we cannot describe z(g)x

using Theorem 3.5.1, for example, because it does not describe the center inintrinsic geometric terms. For that we need to know how the Segal–Sugawaraoperators generating the center transform under changes of coordinates. Wewill address these questions in the next section.

3.5.3 The group of coordinate changes

We start off by doing something which looks coordinate-dependent, but it willeventually allow us to do things in a coordinate-independent way.

Let K denote the complete topological algebra C((t)), and O = C[[t]] withthe induced topology. We study the group of continuous automorphisms ofO, which we denote by AutO. It can be thought of as the group of auto-morphisms of the disc D = SpecO.† As t is a topological generator of thealgebra O, a continuous automorphism of O is determined by its action on t.So we write

ρ(t) = a0 + a1t+ a2t2 + . . . .

In order for this to be an automorphism it is necessary and sufficient thata0 = 0 and a1 6= 0. Note the composition formula in AutO:

ρ(t) φ(t) = φ(ρ(t)).

Let x be a smooth point of a complex curve X. We have the local field Kx

and its ring of integers Ox defined in the previous section. The ring Ox is acomplete local ring, with the maximal ideal mx consisting of those elementsthat vanish at x. By definition, a formal coordinate at x is a topologicalgenerator of mx. We define Autx to be the set of all formal coordinates at x.The group AutO naturally acts on Autx by the formula t 7→ ρ(t). Note thataccording to the above composition formula this is a right action.

Recall that if G is a group, then a G-torsor is a non-empty set S with aright action of G, which is simply transitive. By choosing a point s0 in S,

† Sometimes D is referred to as a “formal disc,” but we prefer the term “disc,” reserving“formal disc” for the formal scheme Spf O (this is explained in detail in Appendix A.1.1.of [FB]).


we obtain an obvious isomorphism between G and S : g 7→ g(s0). However,a priori there is no natural choice of such a point, and hence no naturalisomorphism between the two (just the G-action).

Given a G-torsor S and a representation M of G we can twist M by S bydefining

M = S ×GM.

It is instructive to think of a G-torsor S as a principal G-bundle over a point.Then M is the vector bundle over the point associated to the representationM .

Now observe that the above action of AutO makes Autx into an AutO-torsor. Hence we can twist representations of AutO. The most obviousrepresentations are C[[t]] and C((t)). Not surprisingly, we find that

Ox = Autx ×AutO

C[[t]], Kx = Autx ×AutO

C((t)). (3.5.5)

In other words, we can recover the local field Kx and its ring of integers Ox

from C((t)) and C[[t]] using the action of the group AutO. This gives usa hint as to how to deal with the issue of coordinate-independence that wediscussed in the previous section: if we formulate all our results for the fieldC((t)) keeping track of the action of its group of symmetries AutO, then wecan seamlessly pass from C((t)) to an arbitrary local field Kx by using the“twisting by the torsor” construction. In order to explain this more precisely,we discuss the types of representations of AutO that we should allow.

3.5.4 Action of coordinate changes

Let Aut+O be the subgroup of AutO consisting of those ρ(t) as above thathave a1 = 1. The group AutO is the semi-direct product

AutO = C× n Aut+O,

where C× is realized as the group of rescalings t 7→ at.Denote the Lie algebra of AutO (resp., Aut+O) by Der0O (resp., Der+O).

It is easy to see that

Der0O = tC[[t]]∂t, Der+O = t2C[[t]]∂t.

In particular, this shows that Der+O is a pro-nilpotent Lie algebra and henceAut+O is a pro-unipotent group.†

We wish to describe the category of continuous representations of AutO onvector spaces with discrete topology. As Aut+O is a pro-unipotent group we

† It may appear strange that the Lie algebra of AutO is Der0O and not DerO = C[[t]]∂t.The reason is that the part of DerO spanned by the translation vector field ∂t cannotbe exponentiated within the realm of algebraic groups, but only ind-groups, see [FB],Section 6.2.3, for more details.


know that the exponential map from Der+O to Aut+O is an isomorphism.Hence a discrete representation of Aut+O is the same as a discrete represen-tation of Der+O, i.e., such that for any vector v we have tn∂t · v = 0 for ngreater than a positive integer N .

The Lie algebra of the group C× = t 7→ at of rescalings is Ct∂t ' C.Here we have a discrepancy as the Lie algebra has irreducible representationsparameterized by complex numbers, whereas C× only has representations pa-rameterized by integers: on the representation corresponding to n ∈ Z theelement a ∈ C× acts as an. Furthermore, any linear transformation T on avector space V defines a representation of C, λ 7→ T (in particular, non-trivialextensions are possible because of non-trivial Jordan forms), but any repre-sentation of C× is the direct sum of one-dimensional representations labeledby integers. So the only representations of the Lie algebra C that lift to repre-sentations of C× are those on which the element 1 ∈ C (which corresponds tothe vector field t∂t in our case) acts semi-simply and with integer eigenvalues.

To summarize, a discrete representation of AutO is the same as a discreterepresentation of the Lie algebra Der0O such that the generator t∂t acts semi-simply with integer eigenvalues. In most cases we consider, the action of theLie algebra Der0O may be augmented to an action of DerO = C[[t]]∂t. (Theaction of the additional generator ∂t is important, as it plays the role of aconnection that allows us to identify the objects associated to nearby pointsx, x′ of a curve X.)

In Section 3.1.3 we introduced a topological basis of the Virasoro algebra.The elements Ln = −tn+1 ∂

∂t of this basis with n ≥ 0 (resp., n > 0) generateDer0O (resp., Der+O). The Lie algebra C is generated by L0 = −t∂t. Sup-pose that V is a representation of Der0O, on which L0 acts semi-simply withinteger eigenvalues. Thus, we obtain a Z-grading on V . The commutationrelations in Der0O imply that Ln shifts the grading by −n. Therefore, if theZ-grading is bounded from below, then any vector in V is annihilated by Ln

for sufficiently large n, and so V is a discrete representation of Der0O, whichthen may be exponentiated to AutO.

An example of such a representation is given by the vacuum module Vκ(g).The Lie algebra Der0O (and even the larger Lie algebra DerO) naturally actson it because it acts on g((t)) and preserves the Lie subalgebra g[[t]]⊕C1 fromwhich Vκ(g) is induced. In particular, L0 becomes the vertex algebra gradingoperator, and since this grading takes only non-negative values, we find thatthe action of Der0O may be exponentiated to AutO. The center z(g) is asubspace of Vκc

(g) preserved by AutO, and hence it is a subrepresentation ofVκc

(g).Thus, we can apply the twisting construction to Vκ(g) and z(g). Using

formulas (3.5.5), we find that

Autx ×AutO

Vκ(g) = Vκ(g)x, Autx ×AutO

z(g) = z(g)x,


where Vκ(g)x and z(g)x are defined by formulas (3.5.3) and (3.5.4), respec-tively.

Thus, we have found a way to describe all spaces z(g)x, for different localfields Kx, simultaneously: we simply need to describe the action of DerO onz(g). We will now do this for g = sl2.

3.5.5 Warm-up: Kac–Moody fields as one-forms

Let us start with a simpler example: consider the algebra grVκ(g). As we sawpreviously, it is naturally isomorphic to

Sym(g((t))/g[[t]]) = Fun g∗[[t]]dt.

In Section 3.3.3 we had identified g∗[[t]]dt with g∗[[t]], but now that we wishto formulate our results in a coordinate-independent way, we are not going todo that.

Given a smooth point x of a curve X, we have the disc Dx = SpecOx

and the punctured disc D×x = SpecKx at x. Let ΩOxbe the Ox-module

of differentials. We may think of elements of ΩOxas one-forms on the disc

Dx. Consider the gκ,x-module Vκ(g)x attached to x and its associated gradedgrVκ(g)x. Now observe that

Autx ×AutO

(g∗[[t]]dt) = g∗ ⊗ ΩOx .

Thus, grVκ(g)x is canonically identified with the algebra of functions on thespace of g∗-valued one-forms on Dx.

Now we wish to obtain this result from an explicit computation. Thiscomputation will serve as a model for the computation performed in the nextsection, by which we will identify z(sl2)x with the algebra of functions on thespace of projective connections on Dx.

Let us represent an element φ ∈ g∗[[t]]dt by the formal power series

φ(Ja) = Ja(t) =

∑n<0

Ja

nt−n−1, a = 1, . . . ,dim g.

Then the Ja

n’s are coordinates on g∗[[t]]dt, and we have

Fun g∗[[t]]dt ' C[Ja

n]a=1,...,dim g;n<0.

The Lie algebra DerO acts on Fun g∗[[t]]dt by derivations. Therefore it issufficient to describe its action on the generators J

a

m:

Ln · Ja

m =−mJa

n+m if n+m 6 −10 otherwise

(3.5.6)

We already know from the previous discussion that

Ja(t)dt =

∑m6−1

Ja

mt−m−1dt


is a canonical one-form on the disc D = SpecO. Now we rederive this resultusing formulas (3.5.6). What we need to show is that J

a(t)dt is invariant

under the action of the group AutO of changes of coordinates on D, or,equivalently (since AutO is connected), under the action of the Lie algebraDer0O. This Lie algebra acts in two different ways: it acts on the elementsJ

a

n as in (3.5.6) and it acts on t−n−1dt in the usual way. What we claim isthat these two actions cancel each other.

Indeed, the action of Ln on t−m−1dt is given by

Ln (t−m−1dt) = (m− n)tn−m−1dt.

Therefore

Ln · Ja(t)dt =

∑m6−1

(Ln · Ja

m)t−m−1dt+∑

m6−1

Ja

m(Ln · t−m−1dt)

= −∑

m6−1;n+m6−1

mJa

m+nt−m−1dt+

∑m6−1

Ja

m(m− n)tn−m−1dt

=∑

k6−1

(n− k)Ja

ktn−k−1dt+

∑m6−1

(m− n)Ja

mtn−m−1dt

= 0,

as expected. This implies that for any local field Kx we have a canonicalisomorphism grVκ(g)x ' Fun g∗ ⊗ ΩOx

, but now we have derived it usingnothing but the transformation formula (3.5.6).

Of course, in retrospect our calculation seems tautological. All we saidwas that under the action of AutO the generators J

a

n of grVκ(g) (for fixeda) transform in the same way as tn ∈ K/O. We have the residue pairingbetween K/O = C((t))/C[[t]] and ΩO = C[[t]]dt with respect to which tnn<0

and t−n−1dtn<0 are dual (topological) bases. The expression Ja(t)dt pairs

these two bases. Since the residue pairing is AutO-invariant, it is clear thatJ

a(t) is also AutO-invariant. Thus, the fact that J

a(t) is a canonical one-

form follows immediately once we realize that the generators Ja

n of grVκ(g)transform in the same way as the functions tn under the action of AutO.In the next section we will obtain an analogue of this statement for the Segal–Sugawara operators Sn, the generators of the center z(sl2). This will enableus to give a coordinate-independent description of z(sl2).

3.5.6 The transformation formula for the central elements

From the analysis of the previous section it is clear what we need to do toobtain a coordinate-independent description of z(sl2): we need to find out howthe generators Sn of z(sl2) transform under the action of DerO. A possibleway to do it is to use formula (3.5.2) and the formula

Ln · Jam = −mJa

n+m (3.5.7)


for the action of DerO on the Kac–Moody generators Jan .† This is a rather

cumbersome calculation, but, fortunately, there is an easier way to do it.Let us suppose that κ 6= κc. Then we define the renormalized Segal–

Sugawara operators Sn = κ0κ−κc

Sn, n ≥ −1, as in Section 3.1.1. These areelements of the completed enveloping algebra Uκ(g), which commute with theelements g in the same way as the operators Ln = −tn+1∂t, n ≥ −1; seeformula (3.5.7). Therefore the action of Ln on any other element X of Uκ(g)is equal to [Sn, X].

We are interested in the case when X = Sm. The corresponding commu-tators have already been computed in formula (3.1.3). This formula impliesthe following:

Ln · Sm = [Sn, Sm] = (n−m)Sn+m +n3 − n

12dim(g)

κ

κ0δn,−m.

Though this formula was derived for κ 6= κc, it has a well-defined limit whenκ = κc, and therefore gives us the sought-after transformation formula

Ln · Sm = (n−m)Sn+m +n3 − n

12dim(g)

κc

κ0δn,−m (3.5.8)

(it depends on κ0 because κ0 enters explicitly the definition of Sm).Let us specialize now to the case of sl2. We will use the inner product κ0

defined by the formula κ0(A,B) = TrC2(AB). Then κc = −2κ0, and formula(3.5.8) becomes

Ln · Sm = (n−m)Sn+m −12(n3 − n)δn,−m.

This is the transformation formula for the elements Sm of Uκc(sl2). In par-ticular, it determines the action of the group AutO on the center of Uκc

(sl2)which is topologically generated by the Sm’s. We would like however to un-derstand first the action of AutO on the center z(sl2) of the vertex algebraVκc

(sl2).According to Theorem 3.5.1, we have a homomorphism

C[Sm]m∈Z → z(sl2), X 7→ X|0〉,

and it induces an isomorphism between the quotient of C[Sm]m∈Z by the idealgenerated by Sm,m ≥ −1 and z(sl2). This ideal is preserved by the action ofDerO. Thus, to find the action of DerO on z(sl2) ' C[Sm]m≤−2 we simplyneed to set Sm,m ≥ −1, equal to 0.

† Note the difference between formulas (3.5.6) and (3.5.7). The former describes the actionof DerO on the associated graded algebra gr Vκ(g), whereas the latter describes the action

on Vκ(g) or on eUκ(bg). The two actions are different: for instance, Jan, n ≥ 0, acts by 0

on gr Vκ(g), but Jan, n ≥ 0, act non-trivially on Vκ(g).


This results in the following transformation law:

Ln · Sm =

(n−m)Sn+m if n+m 6 −2− 1

2 (n3 − n) if n+m = 00 otherwise.

(3.5.9)

We now want to work out what sort of geometrical object gives rise to transfor-mation laws like this. It turns out that these objects are projective connectionson the disc D = Spec C[[t]].

3.5.7 Projective connections

A projective connection on D = Spec C[[t]] is a second order differentialoperator

ρ : Ω−1/2O −→ Ω3/2

O (3.5.10)

such that the principal symbol is 1 and the subprincipal symbol is 0.Let us explain what this means. As before, ΩO is the O-module of differ-

entials, i.e., one-forms on D. These look like this: f(t)dt. Now we define theO-module Ωλ

O as the set of “λ-forms”, that is, things which look like this:f(t)(dt)λ. It is a free O-module with one generator, but the action of vectorfields ξ(t)∂t DerO on it depends on λ. To find the transformation formula weapply the transformation t 7→ t+ εξ(t), considered as an element of the groupAutO over the ring of dual numbers C[ε]/(ε2), to f(t)(dt)λ. We find that

f(t)(dt)λ 7→ f(t+ εξ(t))d((t+ εξ(t)))λ.

Now we take the ε-linear term. The result is

ξ(t)∂t · f(t)(dt)λ = (ξ(t)f ′(t) + λf(t)ξ′(t))(dt)λ.

Note, however, that the action of t∂t is semi-simple, but its eigenvaluesare of the form λ + n, n ∈ Z+. Thus, if λ 6∈ Z, the action of DerO cannotbe exponentiated to an action of AutO and we cannot use our twisting con-struction to assign to Ωλ

O a line bundle on a disc Dx = SpecOx (or on anarbitrary smooth algebraic curve for that matter). If λ is a half-integer, forinstance, λ = 1/2, we may construct Ω1/2 by extracting a square root from Ω.On a general curve this is ambiguous: the line bundle we wish to constructis well-defined only up to tensoring with a line bundle L, such that L⊗2 iscanonically trivialized. But on a disc this does not create a problem, as suchL is then trivialized almost canonically; there are two trivializations whichdiffer by a sign, so this is a very mild ambiguity.

In any case, we will not be interested in Ω1/2O itself, but in linear operators

(3.5.10). Once we choose one of the two possible modules Ω1/2O , we take its

inverse as our Ω−1/2O and its third tensor power (over O) as our Ω3/2

O . Thespace of differential operators between them already does not depend on any


choices. The same construction works for an arbitrary curve, so for exampleif X is compact, there are 22g different choices for Ω1/2 (they are called thetacharacteristics), but the corresponding spaces of projective connections arecanonically identified for all of these choices.

Now, a second order differential operator (3.5.10) is something that can bewritten in the form

ρ = v2(t)∂2t + v1(t)∂t + v0(t).

Note here that each vi(t) ∈ C[[t]], but its transformation formula under thegroup AutO is a priori not that of a ordinary function, as we will see below.

The principal symbol of this differential operator is the coefficient v2(t).So the first condition on the projective connection means that ρ is of the form

ρ = ∂2t + v1(t)∂t + v0(t).

The subprincipal symbol of this operator is the coefficient v1(t). The van-ishing of v1(t) means that the operator is of the form

ρ(t) =∂

∂t2− v(t),

where we have set v(t) = −v0(t) for notational convenience.How do projective connections transform under the action of a vector field

ξ(t)∂t ∈ DerO? As we already know the action of ξ(t)∂t on Ω−1/2 and Ω3/2,it is easy to find out how it acts on ρ : Ω−1/2 → Ω3/2. Namely, we find that

ξ(t)∂t ·((∂2

t − v(t)) · f(t)(dt)−1/2)

= ξ(t)∂t ·((f ′′(t)− v(t)f(t))(dt)3/2

)=(ξ(t)(f ′′′(t)− v′(t)f(t)− v(t)f ′(t))

+32(f ′′(t)− v(t)f(t))ξ′(t)

)(dt)3/2

and

(∂2t − v(t)) ·

(ξ(t)∂t · f(t)(dt)−1/2

)

= (∂2t − v(t))

(ξ(t)f ′(t)− 1

2f(t)ξ′(t)

)(dt)−1/2

=(

32ξ′(t)f ′′(t) + ξ(t)f ′′′(t)− v(t)ξ(t)f ′(t)

− 12f(t)ξ′′′(t) +

12v(t)f(t)ξ′(t)

)(dt)3/2.


The action on the projective connection will come from the difference of thesetwo expressions:

−ξ(t)v′(t)− 2v(t)ξ′(t) +12ξ′′′(t).

Thus, under the action of the infinitesimal change of coordinates t 7→ t+ εξ(t)we have

v(t) 7→ v(t) + ε(ξ(t)v′(t) + 2v(t)ξ′(t)− 12ξ′′′(t)). (3.5.11)

It is possible to exponentiate the action of DerO to give the action ofAutO on projective connections. This may done by an explicit, albeit tedious,computation, but one can follow a faster route, as explained in [FB], Section9.2. Suppose that we have two formal coordinates t and s related by theformula t = ϕ(s). Suppose that with respect to the coordinate t the projectiveconnection ρ has the form ∂2

t − v(t). Then with respect to the coordinate s ithas the form ∂2

s − v(s), where

v(s) = v(ϕ(s))ϕ′(s)2 − 12ϕ, s,

and

ϕ, s =ϕ′′′

ϕ′− 3

2

(ϕ′′

ϕ′

)2

(3.5.12)

is the so-called Schwarzian derivative of ϕ.

3.5.8 Back to the center

Now we can say precisely what kinds of geometric objects the Segal–Sugawaraoperators Sm are: the operator Sm behaves as the function on the space ofprojective connections ∂2

t − v(t) picking the t−m−2-coefficient of v(t).

Lemma 3.5.1 The expression

ρS = ∂2t −

∑m6−2

Smt−m−2

defines a canonical projective connection on the disc D = Spec C[[t]], i.e., itis independent of the choice of the coordinate t.

Proof. The proof is similar to the argument used in Section 3.5.5. Weneed to check that the combined action of Ln, n ≥ −1, on ρS , coming from theaction on the Sn’s (given by formula (3.5.9)) and on the projective connection(given by formula (3.5.11)), is equal to 0.


We find that the former is

−∑

m6−2;n+m6−2

(n−m)Sn+mt−m−2 +

∑m6−2;n+m=0

12(n3 − n)t−m−2

= −∑

k6−2

(2n− k)Sktn−k−2 +

12(n3 − n)tn−2,

and the latter is

tn+1v′(t) + 2(n+ 1)tnv(t)− 12(n3 − n)tn−2

=∑

m6−2

(−(m+ 2)Smt

n−m−2 + 2(n+ 1)Smtn−m−2

)− 1

2(n3 − n)tn−2

=∑

m6−2

(2n−m)Smtn−m−2 − 1

2(n3 − n)tn−2.

Summing them gives 0.

Thus, the geometric meaning of the center z(sl2) of the sl2 vertex algebraat the critical level is finally revealed:

The center z(sl2) is isomorphic to the algebra of functions onthe space Proj(D) of projective connections on D.

Applying the twisting by the torsor Autx to Theorem 3.5.1, we obtain thefollowing result.

Theorem 3.5.2 The center z(sl2)x attached to a point x on a curve is canon-ically isomorphic to Fun Proj(Dx), where Proj(Dx) is the space of projectiveconnections on the disc Dx = SpecOx around the point x.

Likewise, we obtain a coordinate-independent description of the centerZ(sl2,x) of the completed universal enveloping algebra Uκ(sl2,x) (this will beproved in Section 4.3.2).

Corollary 3.5.2 The center Z(sl2)x attached to a point x on a curve X iscanonically isomorphic to the algebra Fun Proj(D×x ) on the space of projectiveconnections on the punctured disc D×x = SpecKx around the point x.

We can use Theorem 3.5.2 to construct a family of sl2,κc,x-modules param-eterized by projective connections on Dx. Namely, given ρ ∈ Proj(Dx) welet ρ : z(sl2)x → C be the corresponding character and set

Vρ = Vκc(sl2)/ Im(Ker ρ).

We will see below that all of these modules are irreducible, and these are infact all possible irreducible unramified sl2,κc,x-modules. Thus, we have beenable to link representations of an affine Kac–Moody algebra to projective


connections, which are, as we will see below, special kinds of connections forthe group PGL2, the Langlands dual group of SL2. This is the first step inour quest for the local Langlands correspondence for loop groups.

But what are the analogous structures for an arbitrary affine Kac–Moodyalgebra of critical level? In the next chapter we introduce the notion of op-ers, which are the analogues of projective connections for general simple Liegroups. It will turn out that the center z(g) of Vκc(g) is isomorphic to thealgebra of functions on the space of LG-opers on the disc, where LG is theLanglands dual group to G.

4

Opers and the center for a general Lie algebra

We now wish to generalize Theorem 3.5.2 describing the center z(g) of thevertex algebra Vκc

(g) for g = sl2 to the case of an arbitrary simple Lie algebrag. As the first step, we need to generalize the notion of projective connections,which, as we have seen, are responsible for the center in the case of g = sl2,to the case of an arbitrary g. In Section 4.1 we revisit the notion of projectiveconnection and recast it in terms of flat PGL2-bundles with some additionalstructures. This will enable us to generalize projective connections to thecase of an arbitrary g, in the form of opers. In Section 4.2 we will discuss indetail various definitions and realizations of opers and the action of changesof coordinates on them.

Then we will formulate in Section 4.3 our main result: a canonical isomor-phism between the center z(g) and the algebra of functions on the space ofopers on the disc associated to the Langlands dual group LG. The proof ofthis result occupies the main part of this book. We will use it to describe thecenter Z(g) of the completed enveloping algebra Uκc(g) at the critical level inTheorem 4.3.6. Finally, we will show in Proposition 4.3.9 that the center ofUκ(g) is trivial away from the critical level.

4.1 Projective connections, revisited

In this section we discuss equivalent realizations of projective connections:as projective structures and as PGL2-opers. The last realization is mostimportant to us, as it suggests how to generalize the notion of projectiveconnection (which naturally arose in our description of the center z(g) forg = sl2) to the case of an arbitrary simple Lie algebra g.

4.1.1 Projective structures

The notion of projective connections introduced in Section 3.5.7 makes perfectsense not only on the disc, but also on an arbitrary smooth algebraic curveX over C. By definition, a projective connection on X is a second order

101

102 Opers and the center for a general Lie algebra

differential operator acting between the sheaves of sections of the followingline bundles:

ρ : Ω−1/2X −→ Ω3/2

X , (4.1.1)

such that the principal symbol is 1 and the subprincipal symbol is 0. Here weneed to choose the square root Ω1/2

X of the canonical line bundle ΩX on X.However, as explained in Section 3.5.7, the spaces of projective connectionscorresponding to different choices of Ω1/2

X are canonically identified.We will denote the space of projective connections on X by Proj(X).It is useful to observe that Proj(X) is an affine space modeled on the

vector space H0(X,Ω2X) of quadratic differentials. Indeed, given a projective

connection, i.e., a second order operator ρ as in (4.1.1), and a quadraticdifferential $ ∈ H0(X,Ω2

X), the sum ρ + $ is a new projective connection.Moreover, for any pair of projective connections ρ, ρ′ the difference ρ − ρ′ isa zeroth order differential operator Ω−1/2

X −→ Ω3/2X , which is the same as a

section of Ω2. One can show that for any smooth curve (either compact ornot) the space Proj(X) is non-empty (see [FB], Section 8.2.12). Therefore itis an H0(X,Ω2

X)-torsor.Locally, on an open analytic subset Uα ⊂ X, we may choose a coordinate zα

and trivialize the line bundle Ω1/2. Then, in the same way as in Section 3.5.7,we see that ρα = ρ|Uα may be written with respect to this trivialization asan operator of the form ∂2

zα− vα(zα). In the same way as in Section 3.5.7 it

follows that on the overlap Uα ∩ Uβ , with zα = ϕαβ(zβ), we then have thetransformation formula

vβ(zβ) = vα(ϕαβ(zβ))(∂ϕαβ

∂zβ

)2

− 12ϕαβ , zβ. (4.1.2)

We will now identify projective connections on X with a different kind ofstructures, which we will now define.

A projective chart on X is by definition a covering of X by open subsetsUα, α ∈ A, with local coordinates zα such that the transition functions zβ =fαβ(zα) on the overlaps Uα ∩ Uβ are Mobius transformations

f(z) =az + b

cz + d,

(a b

c d

)∈ PGL2(C).

Two projective charts are called equivalent if their union is also a projectivechart. The equivalence classes of projective charts are called projectivestructures.

Proposition 4.1.1 There is a bijection between the set of projective structureson X and the set of projective connections on X.

Proof. It is easy to check that for a function ϕ(z) the Schwarzian derivativeϕ, z is 0 if and only if ϕ(z) is a Mobius transformation. Hence, given a

4.1 Projective connections, revisited 103

projective structure, we can define a projective connection by assigning thesecond order operator ∂2

zαon each chart Uα. According to formula (4.1.2),

these transform correctly.Conversely, given a projective connection, consider the space of solutions of

the differential equation (∂2

zα− vα

)φ(zα) = 0

on Uα. This has a two-dimensional space of solutions, spanned by φ1,α andφ2,α. Choosing the cover to be fine enough, we may assume that φ2 is never0 and the Wronskian of the two solutions is never 0. Define then

µα =φ1,α

φ2,α: Uα −→ C.

This is well defined and has a non-zero derivative (since the Wronskian isnon-zero). Hence, near the origin it gives a complex coordinate on Uα. In adifferent basis it is clear that the µ’s will be related by a Mobius transforma-tion.

4.1.2 PGL2-opers

We now rephrase slightly the definition of projective structures. This will giveus another way to think about projective connections, which we will be ableto generalize to a more general situation.

We can think of a Mobius transformation as giving us an element of thegroup PGL2(C). Hence, if we have a projective structure on X, then to twocharts Uα and Uβ which overlap we have associated a constant map

fαβ : Uα ∩ Uβ −→ PGL2(C).

As explained in Section 1.2.4, this gives us the structure of a flat PGL2(C)-bundle. Equivalently, as explained in Section 1.2.3, it may be represented as aholomorphic PGL2-bundle F on X with a holomorphic connection ∇ (whichis automatically flat as X is a curve).

However, we have not yet used the fact that we also had coordinates zα

around.The group PGL2(C) acts naturally on the projective line P1. Let us form

the associated P1-bundle

P1F = F ×

PGL2(C)P1

on X. The flat structure on F induces one on P1F . Thus, we have a preferred

system of identifications of nearby fibers of P1F . Now, by definition of F , on

each Uα our bundle becomes trivial, and we can use the coordinate zα todefine a local section of P1

F |Uα= Uα × P1. We simply let the section take

value zα(x) ∈ C ⊂ P1 at the point x.


This is actually a global section of P1F , because the coordinates transform

between each other by exactly the same element of PGL2(C) which we usedas the transition function of our bundle on the overlapping open subsets. Aszα is a coordinate, it has non-vanishing derivative at all points. Therefore thecoordinates are giving us a global section of the P1-bundle P1

F which has anon-vanishing derivative at all points (with respect to the flat connection onP1F ).One may think of this structure as that of a compatible system of local

identifications of our Riemann surface with the projective line. Hence thename “projective structure.”

We now define a PGL2-oper on X to be a flat PGL2(C)-bundle F over Xtogether with a globally defined section of the associated P1-bundle P1

F whichhas a nowhere vanishing derivative with respect to the connection.

So, we have seen that a projective structure, or, equivalently, a projectiveconnection on X, gives rise to a PGL2(C)-oper on X. It is clear that thisidentification is reversible: namely, we use the section to define the localcoordinates; then the flat PGL2-bundle transition functions define the Mobiustransformations of the local coordinates on the overlaps. Hence we see thatprojective structures (and hence projective connections) on X are the samething as PGL2-opers on X.

Now our goal is to generalize the notion of PGL2-oper to the case of an ar-bitrary simple Lie group. This will enable us to give a coordinate-independentdescription of the center of the completed enveloping algebra associated to ageneral affine Kac–Moody algebra gκc .

In order to generalize the concept of oper we need to work out what P1 hasto do with the group PGL2(C) and what the non-vanishing condition on thederivative of the section is telling us.

The group PGL2(C) acts transitively on P1, realized as the variety of linesin C2 = span(e1, e2), and the stabilizer of the line span(e1) is the Borelsubgroup B ⊂ PGL2(C) of upper triangular matrices. Thus, P1 may berepresented as a homogeneous space PGL2(C)/B. The Borel subgroup is aconcept that easily generalizes to other simple Lie groups, so the correspondinghomogeneous space G/B should be the correct generalization of P1.

Now, an element of P1 can be regarded as defining a right coset of B inG. Hence, our section of P1

F gives us above each point x ∈ X a right B-cosetin the fiber Fx. This means that this section gives us a subbundle of theprincipal G-bundle, which is a principal B-bundle.

We recall that a B-reduction of a principal G-bundle F is a B-bundle FB

and an isomorphism F ' FB ×BG. In other words, we are given a subbundle

of F , which is a principal B-bundle such that the action of B on it agreeswith the action restricted from G.

So, a PGL2-oper gives us a flat PGL2-bundle (F ,∇) with a reduction FB

4.1 Projective connections, revisited 105

of F to the Borel subgroup B. The datum of FB is equivalent to the datum ofa section of P1

F . We now have to interpret the non-vanishing condition on thederivative of our section of P1

F that appears in the definition of PGL2-opersin terms of FB . This condition is basically telling us that the flat connectionon F does not preserve the B-subbundle FB anywhere. Let us describe thismore precisely.

As explained in Section 1.2.4, if we choose a local coordinate t on an opensubset U ⊂ X and trivialize the holomorphic PGL2-bundle F , then the con-nection ∇ gives rise to a first order differential operator

∇∂t= ∂t +

(a(t) b(t)c(t) d(t)

), (4.1.3)

where the matrix is in the Lie algebra g = sl2(C), so a(t) + d(t) = 0). Thisoperator is not canonical, but it gets changed by gauge transformations underchanges of trivialization of F (and it also gets transformed under changes ofthe coordinate t).

Our condition on the PGL2-oper may be rephrased as follows: let us choosea local trivialization of F on U which is horizontal with respect to the connec-tion, so ∇∂t = ∂t. Then our section of P1

F may be represented as a functionf : U → P1, and the condition is that its derivative with respect to t has tobe everywhere non-zero.

4.1.3 More on the oper condition

The above description is correct, but not very useful, because in general itis difficult to find a horizontal trivialization explicitly. For that one needs tosolve a system of differential equations, which may not be easy.

It is more practical to choose instead a trivialization of F on U ⊂ X that isinduced by some trivialization of FB on U . Then f becomes a constant mapwhose value is the coset of B in P1 = PGL2/B. But the connection now hasthe form (4.1.3), where the matrix elements are some non-trivial functions.These functions are determined only up to gauge transformations. However,because our trivialization of F is induced by that of FB , the ambiguity con-sists of gauge transformations with values in B only. Any conditions on ∇∂t

that we wish to make now should be invariant under these B-valued gaugetransformations (and coordinate changes of t). For example, connections thatpreserve our B-reduction are precisely the ones in which c(t) = 0; this isclearly a condition that is invariant under B-valued gauge transformations.

Our condition is, to the contrary, that the derivative of our function f :U → P1 (obtained from our section of P1

F using the trivialization of FB) withrespect to ∇∂t

is everywhere non-zero. Note that the value of this derivativeat a point x ∈ U is really a tangent vector to P1 at f(x). Since f(x) = B ∈PGL2/B = P1, the tangent space at f(x) is naturally identified with sl2/b.


Therefore our derivative is represented by the matrix element c(t) appearingin formula (4.1.3). So the condition is simply that c(t) is nowhere vanishing.

Connections satisfying this property may be brought to a standard formby using B-valued gauge transformations. Indeed, first we use the gauge

transformation by the diagonal matrix(c(t)1/2 0

0 c(t)−1/2

)(note that it is

well-defined in PGL2) to bring the connection operator to the form (4.1.3)with c(t) = 1. Next, we follow with the upper triangular transformation(

1 −a0 1

)(∂t +

(a b

1 −a

))(1 a(t)0 1

)= ∂t +

(0 b+ a2 + ∂ta

1 0

).

The result is a connection operator of the form

∇ = ∂t +(

0 v(t)1 0

). (4.1.4)

It is clear from the above construction that there is a unique B-valuedfunction, the product of the above diagonal and upper triangular matrices,that transforms a given operator (4.1.3) with nowhere vanishing c(t) to thisform. Thus, we have used up all of the freedom available in B-valued gaugetransformations in order to bring the connection to this form.

This means that on a small open subset U ⊂ X, with respect to a coordinatet (or on a disc Dx around a point x ∈ X, with respect to a choice of a formalcoordinate t), the space of PGL2-opers is identified with the space of operatorsof the form (4.1.4), where v(t) is a function on U (respectively, v(t) ∈ C[[t]]).

As is well known from the theory of matrix differential equations, the dif-ferential equation ∇Φ(t) = 0 is equivalent to the second order differentialequation (∂2

t − v(t))Φ(t) = 0. Thus, we may identify the space of operswith the space of operators ∂2

t − v(t). These look like projective connections,but to identify them with projective connections we still need to check thatunder changes of coordinates they really transform as differential operatorsΩ−1/2

X −→ Ω3/2X .

To see this, we realize our flat PGL2-bundle F as an equivalence class ofrank two flat vector bundles modulo tensoring with flat line bundles. Usingthe freedom of tensoring with flat line bundles, we may choose a representativein this class such that its determinant is the trivial line bundle with the trivialconnection. We will denote this representative also by F . It is not unique,but may be tensored with any flat line bundle (L,∇) whose square is trivial.The B-reduction FB gives rise to a line subbundle F1 ⊂ F , defined up totensoring with a flat line bundle (L,∇) as above.

A connection ∇ is a map F ⊗TX → F , or equivalently, F → F ⊗ΩX , sinceΩX and TX are dual line bundles. What is the meaning of the oper conditionfrom this point of view? It is precisely that the composition

F1∇−→ F ⊗ ΩX → (F/F1)⊗ ΩX

4.2 Opers for a general simple Lie algebra 107

is an isomorphism of line bundles (it corresponds to the function c(t) appearingin (4.1.3)).

Since we have already identified detF ' F1⊗ (F/F1) with OX , this meansthat (F1)⊗2 ' ΩX , and so F1 ' Ω1/2

X . This also implies that F/F1 ' Ω−1/2X .

Both isomorphisms do not depend on the choice of Ω1/2X as F1 is anyway only

defined modulo tensoring with the square root of the trivial line bundle. Letus pick a particular square root Ω1/2

X . Then we may write

0 −→ Ω1/2X −→ F −→ Ω−1/2

X −→ 0.

Assume for a moment that X is a projective curve of genus g. Then wehave

Ext1(Ω−1/2X ,Ω1/2

X ) ∼= H1(X,ΩX) ∼= H0(X,OX)∗ ∼= C,

so that there may be at most two isomorphism classes of extensions of Ω−1/2X

by Ω1/2X : the split and the non-split ones. If the extension were split, then

there would be an induced connection on the line bundle Ω1/2X . But this

bundle has degree g− 1 and so cannot carry a flat connection if g 6= 1. Thus,for g 6= 1 we have a unique oper bundle corresponding to a non-zero class inH1(X,ΩX). (For g = 1 the two extensions are isomorphic to each other, sothere is again a unique oper bundle.)

For an arbitrary smooth curveX, from the form of the connection we obtainthat if we have a horizontal section of F , then its projection onto Ω−1/2

X is asolution of a second order operator, which is our ∂2

t −v(t). It is also clear thatthis operator acts from Ω−1/2

X to Ω3/2X , and hence we find that it is indeed a

projective connection. Since flat bundles, as well as projective connections,are uniquely determined by their horizontal sections, or solutions, we find thatPGL2-opers X are in one-to-one correspondence with projective connections.

We have made a full circle. We started out with projective connections,have recast them as projective structures, translated that notion into the no-tion of PGL2-opers, which we have now found to correspond naturally toprojective connections by using canonical representatives of the oper connec-tion. Thus, we have found three different incarnations of one and the sameobject: projective connections, projective structures and PGL2-opers. It isthe last notion that we will generalize to an arbitrary simple Lie algebra inplace of sl2.

4.2 Opers for a general simple Lie algebra

In this section we introduce opers associated to an arbitrary simple Lie alge-bra, or equivalently, a simple Lie group of adjoint type. We will see in thenext section that opers are to the center of the vertex algebra Vκc(g) whatprojective connections are to the center of Vκc

(sl2). But there is an important


twist to the story: the center is isomorphic to the algebra of functions on thespace of opers on the disc, associated not to G, but to the Langlands dualgroup LG of G. This appearance of the dual group is very important from thepoint of view of the local Langlands correspondence.

4.2.1 Definition of opers

Let G be a simple algebraic group of adjoint type, B its Borel subgroupand N = [B,B] its unipotent radical, with the corresponding Lie algebrasn ⊂ b ⊂ g.

Thus, g is a simple Lie algebra, and as such it has the Cartan decomposition

g = n− ⊕ h⊕ n+.

We will choose generators e1, . . . , e` (resp., f1, . . . , f`) of n+ (resp., n−). Wehave nαi

= Cei, n−αi= Cfi, see Appendix A.3. We take b = h ⊕ n+ as the

Lie algebra of B. Then n is the Lie algebra of N . In what follows we will usethe notation n for n+.

It will be useful for us to assume that we only make a choice of n = n+

and b = h ⊕ n+, but not of h or n−. We then have an abstract Cartan Liealgebra h = b/n, but no embedding of h into b. We will sometimes choosesuch an embedding, and each such choice then also gives us the lower nilpotentsubalgebra n− (defined as the span of negative root vectors corresponding tothis embedding). Likewise, we will have subgroups N ⊂ B ⊂ G and theabstract Cartan group H = G/B, but no splitting H → B. Whenever wedo use a splitting h → b, or equivalently, H → B, we will explain how thischoice affects our discussion.

Let [n, n]⊥ ⊂ g be the orthogonal complement of [n, n] with respect to anon-degenerate invariant inner product κ0. We have

[n, n]⊥/b '⊕i=1

n−αi .

Clearly, the group B acts on n⊥/b. Our first observation is that there is anopen B-orbit O ⊂ n⊥/b ⊂ g/b, consisting of vectors whose projection on eachsubspace n−αi

is non-zero. This orbit may also be described as the B-orbitof the sum of the projections of the generators fi, i = 1, . . . , `, of any possiblesubalgebra n−, onto g/b. The action of B on O factors through an action ofH = B/N . The latter is simply transitive and makes O into an H-torsor (seeSection 3.5.3).

Let X be a smooth curve and x a point of X. As before, we denote byOx the completed local ring and by Kx its field of fractions. The ring Ox

is isomorphic, but not canonically, to C[[t]]. Then Dx = SpecOx is the discwithout a coordinate and D×x = SpecKx is the corresponding punctured disc.


Suppose now that we are given a principal G-bundle F on a smooth curveX, or Dx, or D×x , together with a connection ∇ (automatically flat) anda reduction FB to the Borel subgroup B of G. Then we define the relativeposition of ∇ and FB (i.e., the failure of ∇ to preserve FB) as follows. Locally,choose any flat connection ∇′ on F preserving FB , and take the difference∇−∇′, which is a section of gFB

⊗ΩX . We project it onto (g/b)FB⊗ΩX . It

is clear that the resulting local section of (g/b)FB⊗ΩX is independent of the

choice ∇′. These sections patch together to define a global (g/b)FB-valued

one-form on X, denoted by ∇/FB .LetX be a smooth curve, orDx, orD×x . Suppose we are given a principalG-

bundle F on X, a connection ∇ on F and a B-reduction FB . We will say thatFB is transversal to∇ if the one-form∇/FB takes values in OFB

⊂ (g/b)FB.

Note that O is C×-invariant, so that O ⊗ ΩX is a well-defined subset of(g/b)FB

⊗ ΩX .Now, a G-oper on X is by definition a triple (F ,∇,FB), where F is a

principal G-bundle F on X, ∇ is a connection on F and FB is a B-reductionof F , such that FB is transversal to ∇.

This definition is due to A. Beilinson and V. Drinfeld [BD1] (in the casewhen X is the punctured disc opers were introduced earlier by V. Drinfeldand V. Sokolov in [DS]).

It is clear that for G = PGL2 we obtain the definition of PGL2-oper fromSection 4.1.2.

4.2.2 Realization as gauge equivalence classes

Equivalently, the transversality condition may be reformulated as saying thatif we choose a local trivialization of FB and a local coordinate t then theconnection will be of the form

∇ = ∂t +∑i=1

ψi(t)fi + v(t), (4.2.1)

where each ψi(t) is a nowhere vanishing function, and v(t) is a b-valued func-tion. One shows this in exactly the same way as we did in Section 4.1.3 inthe case of G = PGL2.

If we change the trivialization of FB , then this operator will get transformedby the corresponding B-valued gauge transformation (see Section 1.2.4). Thisobservation allows us to describe opers on the disc Dx = SpecOx and thepunctured disc D×x = SpecKx in a more explicit way. The same reasoningwill work on any sufficiently small analytic subset U of any curve, equippedwith a local coordinate t, or on a Zariski open subset equipped with an etalecoordinate. For the sake of definiteness, we will consider now the case of thebase Dx.


Let us choose a coordinate t on Dx, i.e., an isomorphism Ox ' C[[t]].Then we identify Dx with D = Spec C[[t]]. The space OpG(D) of G-operson D is the quotient of the space of all operators of the form (4.2.1), whereψi(t) ∈ C[[t]], ψi(0) 6= 0, i = 1, . . . , `, and v(t) ∈ b[[t]], by the action of thegroup B[[t]] of gauge transformations:

g · (∂t +A(t)) = ∂t + gA(t)g−1 − g−1∂tg.

Let us choose a splitting ı : H → B of the homomorphism B → H. ThenB becomes the product B = H n N . The B-orbit O is an H-torsor, andso we can use H-valued gauge transformations to make all functions ψi(t)equal to 1. In other words, there is a unique element of H[[t]], namely, theelement

∏ì=1 ωi(ψi(t)), where ωi : C× → H is the ith fundamental coweight

of G, such that the corresponding gauge transformation brings our connectionoperator to the form

∇ = ∂t +∑i=1

fi + v(t), v(t) ∈ b[[t]]. (4.2.2)

What remains is the group of N -valued gauge transformations. Thus, weobtain that OpG(D) is equal to the quotient of the space OpG(D) of operatorsof the form (4.2.2) by the action of the group N [[t]] by gauge transformations:

OpG(D) = OpG(D)/N [[t]].

This gives us a very concrete realization of the space of opers on the discas gauge equivalence classes.

4.2.3 Action of coordinate changes

In the above formulas we use a particular coordinate t on our disc Dx (and wehave identified Dx with D using this coordinate). As our goal is to formulateall of our results in a coordinate-independent way, we need to figure out howthe gauge equivalence classes introduced above change if we choose anothercoordinate.

So suppose that s is another coordinate on the disc Dx such that t = ϕ(s).In terms of this new coordinate the operator (4.2.2) will become

∇∂t= ∇ϕ′(s)−1∂s

= ϕ′(s)−1∂s +∑i=1

fi + v(ϕ(s)).

Hence we find that

∇∂s = ∂s + ϕ′(s)∑i=1

fi + ϕ′(s) · v(ϕ(s)).

In order to bring it back to the form (4.2.2), we need to apply the gauge


transformation by ρ(ϕ′(s)), where ρ : C× → H is the one-parameter subgroupof H equal to the sum of the fundamental coweights of G, ρ =

∑ì=1 ωi. (Here

we again choose a splitting ı : H → B of the homomorphism B → H). Then,considering ρ as an element of the Lie algebra h = LieH, we have [ρ, ei] = ei

and [ρ, fi] = −fi (see Appendix A.3). Therefore we find that

ρ(ϕ′(s)) ·

(∂s + ϕ′(s)

∑i=1

fi + ϕ′(s) · v(ϕ(s))

)

= ∂s +∑i=1

fi + ϕ′(s)ρ(ϕ′(s)) · v(ϕ(s)) · ρ(ϕ′(s))−1 − ρ(ϕ′′(s)ϕ′(s)

). (4.2.3)

The above formula defines an action of the group AutO on the spaceOpG(D) of opers on the standard disc,

D = Spec C[[t]].

For a discDx around a point x of smooth curveX we may now define OpG(Dx)as the twist of OpG(D) by the AutO-torsor Autx (see Section 3.5.2).

In particular, the above formulas allow us to determine the structure of theH-bundle FH = FB×

BH = FB/N . Let us first describe a general construction

of H-bundles which works on any smooth curve X, or a disc, or a punctureddisc.

Let P be the lattice of cocharacters C× → H. Since G is the group ofadjoint type associated to G, this lattice is naturally identified with the latticeof integral coweights of the Cartan algebra h, spanned by ωi, i = 1, . . . , `.Let P be the lattice of characters H → C×. We have a natural pairing〈·, ·〉 : P × P → Z obtained by composing a character and a cocharacter,which gives us a homomorphism C× → C× that corresponds to an integer.

Given an H-bundle FH on X, for each λ ∈ P we have the associated linebundle Lλ = FH ×

HCλ, where Cλ is the one-dimensional representation of

H corresponding to λ. The datum of FH is equivalent to the data of theline bundles Lλ, λ ∈ P, together with the isomorphisms Lλ ⊗ Lµ ' Lλ+µ

satisfying the obvious associativity condition.Now, given µ ∈ P , we define such data by setting Lλ = Ω〈λ,µ〉

X . Let usdenote the corresponding H-bundle on X by Ωµ. It is nothing but the push-forward of the C×-bundle Ω× with respect to the homomorphism C× → H

given by the cocharacter µ.More concretely, the H-bundle Ωµ may be described as follows: for each

choice of a local coordinate t on an open subset of X we have a trivializationof ΩX generated by the section dt, and hence of each of the line bundlesLλ. If we chose a different coordinate s such that t = ϕ(s), then the twotrivializations differ by the transition function 〈λ, µ(ϕ′(s))〉. In other words,


the transition function for Ωµ is µ(ϕ′(s)). These transition functions give usan alternative way to define the H-bundle Ωµ.

Lemma 4.2.1 The H-bundle FH = FB ×BH = FB/N is isomorphic to Ωρ.

Proof. It follows from formula (4.2.3) for the action of the changes of coor-dinates on opers that if we pass from a coordinate t on Dx to the coordinates such that t = ϕ(s), then we obtain a new trivialization of the H-bundleFH , which is related to the old one by the transition function ρ(ϕ′(s)). Thisprecisely means that FH ' Ωρ.

4.2.4 Canonical representatives

In this section we find canonical representatives in the N [[t]]-gauge classes ofconnections of the form (4.2.2).

In order to do this, we observe that the operator ad ρ defines a gradationon g, called the principal gradation, with respect to which we have a directsum decomposition g =

⊕i gi. In particular, we have b =

⊕i≥0 bi, where

b0 = h.Let now

p−1 =∑i=1

fi.

The operator ad p−1 acts from bi+1 to bi injectively for all i ≥ 0. Hence wecan find for each i ≥ 0 a subspace Vi ⊂ bi, such that bi = [p−1, bi+1]⊕Vi. It iswell-known that Vi 6= 0 if and only if i is an exponent of g, and in that casedimVi is equal to the multiplicity of the exponent i. In particular, V0 = 0.

Let V =⊕

i∈E Vi ⊂ n, where E = d1, . . . , d` is the set of exponents ofg counted with multiplicity. These are exactly the same exponents that wehave encountered previously in Section 2.1.1. They are equal to the orders ofthe generators of the center of U(g) minus 1.

We note that the multiplicity of each exponent is equal to 1 in all casesexcept the case g = D2n, dn = 2n, when it is equal to 2.

Lemma 4.2.2 ([DS]) The action of N [[t]] on OpG(D) is free.

Proof. We claim that each element of ∂t + p−1 + v(t) ∈ OpG(D) can beuniquely represented in the form

∂t + p−1 + v(t) = exp (adU) · (∂t + p−1 + c(t)) , (4.2.4)

where U ∈ n[[t]] and c(t) ∈ V [[t]]. To see this, we decompose with respect tothe principal gradation: U =

∑j≥0 Uj , v(t) =

∑j≥0 vj(t), c(t) =

∑j∈E cj(t).

Equating the homogeneous components of degree j in both sides of (4.2.4),


we obtain that ci + [Ui+1, p−1] is expressed in terms of vi, cj , j < i, andUj , j ≤ i. The injectivity of ad p−1 then allows us to determine uniquely ci

and Ui+1. Hence U and c satisfying equation (4.2.4) may be found uniquelyby induction, and the lemma follows.

There is a special choice of the transversal subspace V =⊕

i∈E Vi. Namely,there exists a unique element p1 in n, such that p−1, 2ρ, p1 is an sl2-triple.This means that they have the same relations as the generators e, h, f ofsl2 (see Section 2.1.1).

For example, for g = sln (with the rank ` = n− 1) we have

p−1 =

0 . . .

1 01 0

. . . . . .1 0

,

p0 =

n− 1

n− 3n− 5

. . .−n+ 1

,

p1 =

0 1(n− 1)0 2(n− 2)

0 3(n− 3). . . . . .

(n− 1)10

.

In general, we have p1 =∑`

i=1miei, where the ei’s are generators of n+

and the mi’s are certain coefficients uniquely determined by the conditionthat p−1, 2ρ, p1 is an sl2-triple.

Let V can =⊕

i∈E Vcani be the space of ad p1-invariants in n. Then p1 spans

V can1 . Let pj be a linear generator of V can

dj. If the multiplicity of dj is greater

than 1, then we choose linearly independent vectors in V candj

.In the case of g = sln we may choose as the elements pj , j = 1, . . . , n − 1,

the matrices pj1 (with respect to the matrix product).

According to Lemma 4.2.2, each G-oper may be represented by a uniqueoperator ∇ = ∂t + p−1 + v(t), where v(t) ∈ V can[[t]], so that we can write

v(t) =∑j=1

vj(t) · pj , vj(t) ∈ C[[t]].


Let us find out how the group of coordinate changes acts on the canonicalrepresentatives.

Suppose now that t = ϕ(s), where s is another coordinate on Dx suchthat t = ϕ(s). With respect to the new coordinate s, ∇ becomes equal to∂s + v(s), where v(s) is expressed via v(t) and ϕ(s) as in formula (4.2.3).By Lemma 4.2.2, there exists a unique operator ∂s + p−1 + v(s) with v(s) ∈V can[[s]] and g ∈ B[[s]], such that

∂s + p−1 + v(s) = g · (∂s + v(s)) . (4.2.5)

It is straightforward to find that

g = exp(

12ϕ′′

ϕ′· p1

)ρ(ϕ′), (4.2.6)

v1(s) = v1(ϕ(s)) (ϕ′)2 − 12ϕ, s, (4.2.7)

vj(s) = vj(ϕ(s)) (ϕ′)dj+1, j > 1, (4.2.8)

where ϕ, s is the Schwarzian derivative (3.5.12).Formula (4.2.6) may be used to describe the B-bundle FB . Namely, in

Section 4.1.3 we identified the PGL2 bundle FPGL2 underlying all PGL2-opers and its BPGL2-reduction FBP GL2

. Let ı : PGL2 → G be the principalembedding corresponding to the principal sl2 subalgebra of g that we havebeen using (recall that by our assumption G is of adjoint type). We denoteby ıB the corresponding embedding BPGL2 → B. Then, according to for-mula (4.2.6), the G-bundle F underlying all G-opers and its B-reduction FB

are isomorphic to the bundles induced from FPGL2 and FBP GL2under the

embeddings ı and ıB , respectively.Formulas (4.2.7) and (4.2.8) show that under changes of coordinates, v1

transforms as a projective connection, and vj , j > 1, transforms as a (dj +1)-differential on Dx. Thus, we obtain an isomorphism

OpG(Dx) ' Proj(Dx)×⊕j=2

Ω⊗(dj+1)Ox

, (4.2.9)

where Ω⊗nOx

is the space of n-differentials on Dx and Proj(Dx) is the Ω⊗2Ox

-torsor of projective connections on Dx.

This analysis carries over verbatim to the case of the punctured disc D×x .In particular, we obtain that for each choice of coordinate t on Dx we havean identification between OpG(D×x ) with the space of operators ∇ = ∂t +p−1 + v(t), where now v(t) ∈ V can((t)). The action of changes of coordinatesis given by the same formula as above, and so we obtain an analogue of theisomorphism (4.2.9):

OpG(D×x ) ' Proj(D×x )×⊕j=2

Ω⊗(dj+1)Kx

. (4.2.10)


The natural embedding OpG(Dx) → OpG(D×x ) is compatible with the naturalembeddings of the right hand sides of (4.2.9) and (4.2.10).

In the same way we obtain an identification for a general smooth curve X:

OpG(X) ' Proj(X)×⊕j=2

Γ(X,Ω⊗(dj+1)X ).

Just how canonical is this “canonical form”? Suppose we do not wish tomake any choices other than that of a Borel subalgebra b in g. The definition ofthe canonical form involves the choice of the sl2-triple p−1, 2ρ, p1. However,choosing such an sl2-triple is equivalent to choosing a splitting h → b of theprojection b/n → h and choosing the generators of the corresponding one-dimensional subspaces n−αi . It is easy to see that the group B acts simplytransitively on the set of these choices, and hence on the set of sl2-triples.

Given an sl2-triple p−1, 2ρ, p1, we also chose basis elements pi, i ∈ E, ofV can, homogeneous with respect to the grading defined by ρ. This allowed usto make the identification (4.2.9). However, even if the multiplicity of di isequal to one, these elements are only well-defined up to a scalar. Therefore,if we want a truly canonical form, we should consider instead of the vectorspace

⊕ì=2 Ωdi

Ox· pi that we used in (4.2.9), the space

V can>1,x = Γ(Dx,Ω× ×

C×V can

>1 (1)),

where V can>1 (1) is the vector space

⊕i>0 V

cani with the grading shifted by 1,

which no longer depends on the choice of a basis. Then we obtain a morecanonical identification

OpG(Dx) ' Proj(Dx)× V can>1,x, (4.2.11)

and similarly for the case of D×x and a more general curve X, which nowdepends only on the choice of sl2-triple. In particular, since Proj(Dx) isan Ω⊗2

Ox-torsor, we find, by identifying Ω⊗2

Ox= Ω2

Ox· p1, that OpG(Dx) is a

V canx -torsor, where

V canx = Γ(Dx,Ω× ×

C×V can(1)).

Suppose now that we have another sl2-triple p−1, 2˜ρ, p1, and the corre-sponding space V can. Then both are obtained from the original ones by theadjoint action of a uniquely determined element b ∈ B. Given an oper ρ, weobtain its two different canonical forms:

∇ = ∂t + p−1 + v1(t)p1 + v>1(t), ∇ = ∂t + p−1 + v1(t)p1 + v>1(t),

where v(t) ∈ V can>1 and v(t) ∈ V can

>1 . But then it follows from the constructionthat we have ∇ = b∇b−1, which means that v1(t) = v1(t) and

v>1(t) = bv>1(t)b−1.


Since all possible subspaces V can>1 are canonically identified with each other

in this way, we may identify all of them with a unique “abstract” gradedvector space V abs

<1 (note however that V abs<1 cannot be canonically identified

with a subspace of b). We define its twist V abs>1,x in the same way as above.

Then the “true” canonical form of opers is the identification

OpG(Dx) ' Proj(Dx)× V abs>1,x, (4.2.12)

which does not depend on any choices (and similarly for the case of D×x anda more general curve X). In particular, we find, as above, that OpG(Dx) is aV abs

x -torsor, where V absx is the twist of the abstract graded space V abs defined

as before.

4.2.5 Alternative choice of representatives for sln

In the case of g = sln there is another choice of representatives of oper gaugeclasses which is useful in applications. Namely, we choose as the transversalsubspace V =

⊕n−1i=1 Vi the space of traceless matrices with non-zero entries

only in the first row (so its ith component Vi is the space spanned by thematrix E1,i+1).

The corresponding representatives of PGLn-opers have the form

∂t +

0 u1 u2 · · · un−1

1 0 0 · · · 00 1 0 · · · 0...

. . . . . . · · ·...

0 0 · · · 1 0

. (4.2.13)

In the same way as in the case of sl2 one shows that this space coincides withthe space of nth order differential operators

∂nt − u1(t)∂n−2

t + . . .+ un−2(t)∂t − (−1)nun−1(t) (4.2.14)

acting from Ω−(n−1)/2 to Ω(n+1)/2. They have principal symbol 1 and sub-principal symbol 0.

The advantage of this form is that it is very explicit and gives a concreterealization of PGLn-opers in terms of scalar differential operators. There aresimilar realizations for other classical Lie groups (see [DS, BD3]). The disad-vantage is that this form is not as canonical, because the “first row” subspaceis not canonically defined. In addition, the coefficients ui(t) of the operator(4.2.14) transform in a complicated way under changes of coordinates, unlikethe coefficients vi(t) of the canonical form from the previous section.

4.3 The center for an arbitrary affine Kac–Moody algebra 117

4.3 The center for an arbitrary affine Kac–Moody algebra

We now state one of the main results of this book: a coordinate-independentdescription of the center z(g) of the vertex algebra Vκc

(g). This description,given in Theorems 4.3.1 and 4.3.2, generalizes the description of z(sl2) givenin Theorem 3.5.2. We will use it to describe the center Z(g) of the completedenveloping algebra Uκc

(g) at the critical level in Theorem 4.3.6. Finally, wewill show in Proposition 4.3.9 that the center of Uκ(g) is trivial away fromthe critical level.

4.3.1 The center of the vertex algebra

Let us recall from Section 3.5.2 that for any smooth point x of a curve X wehave the affine Kac–Moody algebra gx associated to x and the algebra z(g)x

defined by formula (3.5.4).Let LG be the simple Lie group of adjoint type associated to the Langlands

dual Lie algebra Lg of g. By definition, Lg is the Lie algebra whose Cartanmatrix is the transpose of the Cartan matrix of g (see Appendix A.3). Thus,LG is the Langlands dual group (as defined in Section 1.1.5) of the groupG, which is the connected simply-connected simple Lie group with the Liealgebra g.

The following theorem proved by B. Feigin and myself [FF6] (see also [F4])is the central result of this book. The detailed proof of this result will bepresented in the following chapters.

Theorem 4.3.1 The algebra z(g)x is naturally isomorphic to the algebra offunctions on the space OpLG(Dx) of LG-opers on the disc Dx.

Equivalently, we may consider the algebra z(g) corresponding to the stan-dard disc D = SpecO,O = C[[t]], but keep track of the action of the groupAutO and the Lie algebra DerO. They act on z(g) ⊂ Vκc(g) and on the spaceOpLG(D) (and hence on the algebra Fun OpLG(D) of functions on it).

Theorem 4.3.2 The center z(g) is isomorphic to the algebra FunOpLG(D)in a (DerO,AutO)-equivariant way.

Using the isomorphism (4.2.9), we obtain an isomorphism

OpLG(D) ' C[[t]]⊕`

(of course, it depends on the coordinate t). Therefore each oper is representedby an `-tuple of formal Taylor series (v1(t), . . . , v`(t)). Let us write

vi(t) =∑n<0

vi,nt−n−1.


Then we obtain an isomorphism

FunOpLG(D) ' C[vi,n]i=1,...,`;n<0. (4.3.1)

Let Si, i = 1, . . . , `, be the elements of z(g) corresponding to

vi,−1 ∈ FunOpLG(D)

under the isomorphism of Theorem 4.3.2. Let us use the fact that it is equiv-ariant under the action of the operator L−1 = −∂t ∈ DerO. On the sideof FunOpLG(D) it is clear that 1

m! (−∂t)mvi,−1 = vi,−m−1. Hence we obtainthat under the isomorphism of Theorem 4.3.2 the generators vi,−m−1 go to

1m!Lm−1Si =

1m!TmSi,

because L−1 coincides with the translation operator T on the vertex algebraVκc

(g). This implies that

vi,−m−1 7→ Si,(−m−1)|0〉,

where the Si,(n)’s are the Fourier coefficients of the vertex operator

Y (Si, z) =∑n<0

Si,(n)z−n−1.

Indeed, we recall that z(g) is a commutative vertex algebra, and so we can useformula (2.2.1). Note that the operators Si,(n), n ≥ 0, annihilate the vacuumvector |0〉, and because they commute with the action of gκc , their action onVκc

(g) is identically equal to zero.Furthermore, we obtain that any polynomial in the vi,n’s goes under the

isomorphism of Theorem 4.3.2 to the corresponding polynomial in the Si,(n)’s,applied to the vacuum vector |0〉 ∈ Vκc

(g). Thus, we obtain that

z(g) = C[Si,(n)]i=1,...,`;n<0|0〉. (4.3.2)

This identification means, in particular, that the map gr z(g) → Inv g∗[[t]]from Lemma 3.3.1 is an isomorphism.†

To see that, we recall the description of the algebra Inv g∗[[t]] from Theo-rem 3.4.2:

Inv g∗[[t]] = C[P i,n]i=1,...,`;n<0,

where P i, i = 1, . . . , `, are homogeneous generators of the algebra Inv g∗ of de-grees di +1, i = 1, . . . , `. To compare the two spaces, gr z(g) and Inv g∗[[t]], wecompute their formal characters, i.e., the generating function of their gradeddimensions with respect to the grading operator L0 = −t∂t.

† We remark that this is, in fact, one of the steps in our proof of Theorem 4.3.2 givenbelow.


If V is a Z-graded vector space V =⊕

n∈Z Vn with finite-dimensional gradedcomponents, we define its formal character as

chV =∑n∈Z

dimVnqn.

Now, it follows from the definition of the generators P i,n of Inv g∗[[t]] givenin Section 3.3.4 that degP i,m = di −m. Therefore we find that

ch Inv g∗[[t]] =∏i=1

∏ni≥di+1

(1− qni)−1. (4.3.3)

On the other hand, from the description of OpLG(D) given by formula(4.2.9) we obtain the action of −t∂t ∈ DerO on the generators vi,m. We findthat deg vi,m = di −m. Hence the character of FunOpLG(D) (and thereforethe character of z(g)) coincides with the right hand side of formula (4.3.3).Since we have an injective map gr z(g) → Inv g∗[[t]] of two graded spaceswhose characters coincide, we obtain the following:

Proposition 4.3.3 The natural embedding gr z(g) → Inv g∗[[t]] is an isomor-phism.

In other words, the center z(g) is “as large as possible.” This means that thegenerators P i,−1 of the algebra Inv g∗[[t]] may be “lifted” to g[[t]]-invariantvectors Si ∈ z(g), i = 1, . . . , `. In other words, P i,−1 is the symbol of Si. Itthen follows that P i,n is the symbol of Si,(n)|0〉 ∈ z(g) for all n < 0.

In particular, P 1,−1 is the quadratic Casimir generator given by the formula

P 1,−1 =12J

a

−1Ja,−1,

up to a non-zero scalar. Its lifting to z(g) is the Segal–Sugawara vector S1

given by formula (3.1.1). Therefore the generators S1,(n), n < 0, are theSegal–Sugawara operators Sn−1.

It is natural to ask whether it is possible to find explicit liftings to z(g)of other generators P i,−1, i = 1, . . . , `, of Inv g∗[[t]]. So far, the answer isknown only in special cases. For g = sln explicit formulas for P i,−1 may beconstructed using the results of R. Goodman and N. Wallach [GW] whichrely on some intricate invariant theory computations.† For g of classical typesA`, B`, C` another approach was suggested by T. Hayashi in [Ha]: he con-structed explicitly the next central elements after the Segal–Sugawara oper-ators, of degree 3 in the case of A`, and of degree 4 in the case of B` andC`. He then generated central elements of higher degrees by using the Pois-son structure on the center of Uκc

(g) discussed in Section 8.3.1 below (which

† Recently, an elegant formula of a different kind for g = sln was obtained in [CT].


Hayashi had essentially introduced for this purpose). Unfortunately, this ap-proach does not not give the entire center when g = D` (as the Pfaffianappears to be out of reach) and when g is an exceptional Lie algebra. For g

of types A`, B`, C` one can prove in this way the isomorphism (4.3.2), but notTheorem 4.3.2 identifying the center z(g) with the algebra FunOpLG(D).

The proof of Theorem 4.3.2 given by B. Feigin and myself [FF6, F4], which ispresented in this book, does not rely on explicit formulas for the generators ofthe center. Instead, we identify z(g) and FunOpLG(D) as two subalgebras ofanother algebra, namely, the algebra of functions on the space of connectionson a certain LH-bundle on the disc, where LH is a Cartan subgroup of LG.

4.3.2 The center of the enveloping algebra

We now use the above description of the center of z(g) to describe the centerZ(g) of the completed enveloping algebra Uκc

(g) at the critical level (seeSection 2.1.2 for the definition).

Let B ∈ z(g) ⊂ Vκc(g). Then g[[t]] · B = 0 and formula (3.2.2) for thecommutation relations imply that all elements B[k] ∈ Uκc

(g), as definedin Section 3.2.1, commute with the entire affine Kac–Moody algebra gκc

.Therefore they are central elements of Uκc

(g). Moreover, any element ofU(z(g)) ⊂ U(Vκc(g)) = Uκc(g) is also central (here we use the enveloping al-gebra functor U defined in Section 3.2.3 and Lemma 3.2.2). Thus, we obtaina homomorphism U(z(g))→ Z(g).

Now, the algebra U(z(g)) is a completion of a polynomial algebra. Indeed,recall from the previous section that we have elements Si, i = 1, . . . , `, in z(g)which give us the isomorphism (4.3.2). Let Si,[n], n ∈ Z, be the elementsof Uκc(g) corresponding to Si ∈ Vκc(g) (see Section 3.2.1). It follows fromthe definition that the topological algebra U(z(g)) is the completion of thepolynomial algebra C[Si,[n]]i=1,...,`;n∈Z with respect to the topology in whichthe basis of open neighborhoods of 0 is formed by the ideals generated bySi,[n], i = 1, . . . , `;n > N + di + 1, for N ∈ Z+. This topology is equivalentto the topology in which the condition n > N + di + 1 is replaced by thecondition n > N .

Thus, the completed polynomial algebra U(z(g)) maps to the center Z(g)of Uκc

(g).

Proposition 4.3.4 This map is an isomorphism, and so Z(g) is equal toU(z(g)).

Proof. We follow the proof given in [BD1], Theorem 3.7.7.We start by describing the associated graded algebra of Z(g). The Poincare–

Birkhoff–Witt filtration on Uκc(g) induces one on the completion Uκc(g).The associated graded algebra grUκc

(g) (recall the definition given in Sec-


tion 3.2.3) is the algebra Sym g((t)) =⊕

i≥0 Symi g((t)). Let IN be the ideal inSym g((t)) generated by g⊗ tNC[[t]]. The associated graded algebra gr Uκc(g)

is the completion Sym g((t)) =⊕

i≥0 Symig((t)) of Sym g((t)), where

Symig((t)) = lim

←−Symi g((t))/(IN ∩ Symi g((t))).

In other words, Symig((t)) is the space Funi(g∗ ⊗ t−NC[[t]]) of polynomial

functions on g((t)) ' g∗((t))dt ' g∗((t)) of degree i. By such a function we meana function on g∗((t)) such that for each N ∈ Z+ its restriction to g∗⊗t−NC[[t]]is a polynomial function of degree i (i.e., it comes by pull-back from a poly-nomial function on a finite-dimensional vector space g∗ ⊗ (t−NC[[t]]/tnC[[t]])for sufficiently large n).

The Lie algebra g((t)) naturally acts on each space Symig((t)) via the adjoint

action, and its Lie subalgebra g[[t]] preserves the subspace IN ∩ Symi g((t)) ofthose functions which vanish on g⊗ tNC[[t]].

Let Invg∗((t)) =⊕

i≥0 Invig((t)) be the subalgebra of g((t))-invariant ele-

ments of Sym g((t)). For each N ∈ Z+ we have a surjective homomorphismobtained by taking the quotient by IN :

Symig((t))→ Funi(g∗ ⊗ t−NC[[t]]).

The image of Invig∗((t)) under this homomorphism is contained in the space

of g[[t]]-invariants in Funi(g∗ ⊗ t−NC[[t]]).But we have a natural isomorphism

Fun(g∗ ⊗ t−NC[[t]]) ' Fun(g∗[[t]])

(multiplication by tN ) which commutes with the action of g[[t]]. Since weknow the algebra of g[[t]]-invariant functions on g∗[[t]] from Theorem 3.4.2,we find the algebra of g[[t]]-invariant functions on g∗⊗ t−NC[[t]] by using thisisomorphism. To describe it, let J

a

n be a linear functional on g∗((t)) defined asin formula (3.3.4). Note that the restriction of J

a

n, n ≥ N , to g∗ ⊗ t−NC[[t]]is equal to 0. Let us write

P i(z) = P i(Ja(z)) =

∑n∈Z

P i,nz−n−1,

where

Ja(z) =

∑n∈Z

Ja

nz−n−1.

Now we obtain from the above isomorphism that the algebra of g[[t]]-invariantfunctions on g∗⊗ t−NC[[t]] is the free polynomial algebra with the generatorsP i,n, n < (di + 1)N .


However, one checks explicitly that each P i,n is not only g[[t]]-invariant,but g((t))-invariant. This means that the homomorphism

Inv g∗((t))→ (Fun(g∗ ⊗ t−NC[[t]]))g[[t]]

is surjective for all N ∈ Z+. Hence it follows that Inv g∗((t)) is the inverselimit of the algebras (Fun(g∗ ⊗ t−NC[[t]]))g[[t]].

Therefore Inv g∗((t)), is the completion of the polynomial algebra

C[P i,n]i=1,...,`;n∈Z

with respect to the topology in which the basis of open neighborhoods of 0is formed by the subspaces of the polynomials of fixed degree that lie in theideals generated by P i,n, with n > N , for N ∈ Z+ (using the last condition isequivalent to using the condition n ≥ (di + 1)N , which comes from the aboveanalysis).

Now we are ready to prove the proposition. Recall the definition of Uκc(g) asthe inverse limit of the quotients of Uκc

(g)/IN by the left ideals IN generatedby g⊗ tNC[[t]]. Therefore

Z(g) = lim←−

Z(g)/Z(g) ∩ IN .

We have an injective map Z(g)/Z(g) ∩ IN → (Uκc(g)/IN )g[[t]]. In addition,we have an injective map

gr((Uκc(g)/IN )g[[t]]) → (grUκc(g)/IN )g[[t]] = (Fun(g∗ ⊗ t−NC[[t]]))g[[t]].

We already know that C[Si,[n]]i=1,...,`;n∈Z is a subalgebra of Z(g). By con-struction, the symbol of Si,[n] is equal to P i,n. Therefore the image ofC[Si,[n]]i=1,...,`;n∈Z under the composition of the above maps is the entire(Fun(g∗ ⊗ t−NC[[t]]))g[[t]].

This implies that all the intermediate maps are isomorphisms. In particular,we find that for each N ∈ Z+ we have

Z(g)/Z(g) ∩ IN = C[Si,[ni]]i=1,...,`;ni≤N(di+1).

Therefore the center Z(g) is equal to the completion of the polynomial algebraC[Si,[n]]i=1,...,`;n∈Z with respect to the topology in which the basis of openneighborhoods of 0 is formed by the ideals generated by Si,[n], i = 1, . . . , `;n >N . But this is precisely U(z(g)). This completes the proof.

As a corollary we obtain that Z(g) is isomorphic to U(FunOpLG(D)).

Lemma 4.3.5 The algebra U(FunOpLG(D)) is canonically isomorphic to thetopological algebra FunOpLG(D×) of functions on the space of LG-opers onthe punctured disc D× = Spec C((t)).


Proof. As explained in Section 4.3.1, the algebra FunOpLG(D) is iso-morphic to the polynomial algebra C[vi,n]i=1,...,`;n<0, where the vi,n’s are thecoefficients of the oper connection

∇ = ∂t + p−1 +∑i=1

vi(t)pi, (4.3.4)

where vi(t) =∑

n<0 vi,nt−n−1. The construction of the functor U implies

that the algebra U(OpLG(D)κ∨0) is the completion of the polynomial algebra

in the variables vi,n, i = 1, . . . , `;n ∈ Z, with respect to the topology in whichthe base of open neighborhoods of 0 is formed by the ideals generated byvi,n, n > N . But this is precisely the algebra of functions on the space ofopers of the form (4.3.4), where vi(t) =

∑n∈Z vi,nt

−n−1 ∈ C((t)).

Now Theorem 4.3.2, Proposition 4.3.4 and Lemma 4.3.5 imply the following:

Theorem 4.3.6 The center Z(g) is isomorphic to the algebra FunOpLG(D×)in a (DerO,AutO)-equivariant way.

We will see later that there are other conditions that this isomorphismsatisfies, which fix it almost uniquely.

Using this theorem, we can describe the center Z(gx) of the envelopingalgebra Uκc

(g)x of the Lie algebra gκc,x.

Corollary 4.3.7 The center Z(gx) is isomorphic to the algebra FunOpLG(D×x )of functions on the space of LG-opers on D×x .

We also want to record the following useful result, whose proof is borrowedfrom [BD1], Remark 3.7.11(iii). Note that the group G((t)) acts naturally onUκc(g).

Proposition 4.3.8 The action of G((t)) on Z(g) ⊂ Uκc(g) is trivial.

Proof. Let Uκc(g)≤i be the ith term of the filtration on Uκc(g) induced bythe PBW filtration on Uκc

(g) (see the proof of Proposition 4.3.4). It is clearthat the union of Uκc

(g)≤i, i ≥ 0, is dense in Uκc(g). Moreover, it follows fromthe description of the center as a completion of the algebra C[Si,[n]]i=1,...,`;n∈Z

that the union of Z(g) ∩ Uκc(g)≤i, i ≥ 0, is dense in Z(g). Therefore it is

sufficient to prove that G((t)) acts trivially on Z(g) ∩ Uκc(g) (clearly, G((t))

preserves each term Uκc(g)≤i).

In Proposition 4.3.4 we described the associated graded algebra grZ(g) ofZ(g) with respect to the above filtration. Namely, grZ(g) is a completionof a polynomial algebra. Using explicit formulas for the generators P i,n ofthis algebra presented above, we find that G((t)) acts trivially on grZ(g).Therefore the action of G((t)) on Z(g) ∩ Uκc(g) factors through a unipotent


algebraic group U . But then the corresponding action of its Lie algebra g((t))factors through an action of the Lie algebra of U . From our assumption thatG is simply-connected we obtain that G((t)) is connected. Therefore U is alsoconnected.

Clearly, the differential of a non-trivial action of a connected unipotentLie group is also non-trivial. Therefore we find that if the action of G((t)) onZ(g)∩Uκc

(g) is non-trivial, then the action of its Lie algebra g((t)) is also non-trivial. But it is obvious that g((t)) acts by 0 on Z(g). Therefore the actionof G((t)) on Z(g) ∩ Uκc

(g), and hence on Z(g), is trivial. This completes theproof.

Note that if we had not assumed that G were simply-connected, then thefundamental group of G would be a finite group Γ, which would coincidewith the group of components of G((t)). The above argument shows that theaction of G((t)) factors through Γ. But a finite group cannot have non-trivialunipotent representations. Hence we obtain that G((t)) acts trivially on Z(g)even without the assumption that G is simply-connected.

4.3.3 The center away from the critical level

The above results describe the center of the vertex algebra Vκ(g) and thecenter of the completed enveloping algebra Uκ(g) at the critical level κ = κc.But what happens away from the critical level?

In Proposition 3.3.3 we have answered this question in the case of the vertexalgebra: it is trivial, i.e., consists of scalars only. Now, for completeness, weanswer it for the completed enveloping algebra Uκ(g) and show that it is trivialas well. Therefore we are not missing anything, as far as the center of Uκ(g)is concerned, outside of the critical level.

Proposition 4.3.9 The center of Uκ(g) consists of the scalars for κ 6= κc.

Proof. Consider the Lie algebra g′κ = Cd n gκ, where d is the operatorwhose commutation relations with gκ correspond to the action of the vectorfield t∂t. Let U ′κ(g) be the quotient of the universal enveloping algebra of g′κby the ideal generated by 1− 1, and U ′κ(g) its completion defined in the sameway as Uκ(g).

Now, the algebra Uκ(g), κ 6= κc, contains the operator L0, which commuteswith gκ as −d. Therefore L0 + d is a central element of U ′κ(g), and Uκ(g) isisomorphic to the quotient of U ′κ(g) by the ideal generated by L0 + d.

Let Zκ(g) be the center of Uκ(g) and Z ′κ(g) the center of U ′κ(g). We have anatural embedding Uκ(g)→ U ′κ(g). Since any element A of Zκ(g) must satisfy[L0, A] = 0, we find that the image of A in U ′κ(g) is also a central element.Therefore we find that Z ′κ(g) = Zκ(g)⊗C C[L0 + d].

The description of the center Z ′κ(g) follows from the results of V. Kac


[K1]. According to Corollary 1 of [K1], for κ 6= κc there is an isomorphismbetween Z ′κ(g) and the algebra of invariant functions on the hyperplane (h)∗κ,corresponding to level κ, in the dual space to the extended Cartan subalgebrah = (h⊗1)⊕C1⊕Cd, with respect to the (ρ-shifted) action of the affine Weylgroup. But it is easy to see that the polynomial functions on (h)∗κ satisfyingthis condition are spanned by the powers of the polynomial on (h)∗κ obtainedby restriction of the invariant inner product on h∗. This quadratic polynomialgives rise to the central element L0+d. Therefore Z ′κ(g) = Zκ(g)⊗C C[L0+d],and so Zκ(g) = C.

5

Free field realization

We now set out to prove Theorem 4.3.2 establishing an isomorphism betweenthe center of the vertex algebra Vκc

(g) and the algebra of functions on thespace of LG-opers on the disc. An essential role in this proof is played bythe Wakimoto modules over gκc . Our immediate goal is to construct thesemodules and to study their properties. This will be done in this chapter andthe next, following [FF4, F4].

5.1 Overview

Our ultimate goal is to prove that the center of the vertex algebra Vκc(g)

is isomorphic to the algebra of functions on the space of LG-opers on thedisc. It is instructive to look first at the proof of the analogous statementin the finite-dimensional case, which is Theorem 2.1.1. The most direct wayto describe the center Z(g) of U(g) is to use the so-called Harish-Chandrahomomorphism Z(g) → Fun h∗, where h is the Cartan subalgebra of g, andto prove that its image is equal to the subalgebra (Fun h∗)W of W -invariantfunctions, where W is the Weyl group acting on h∗ (see Appendix A.3). Howcan one construct this homomorphism? One possible way is to use geometry;namely, the infinitesimal action of the Lie algebra g on the flag manifoldFl = G/B−, where B− is the lower Borel subgroup of G. This action preservesthe open dense B+-orbit U (where B+ is the upper Borel subgroup of G)in Fl, namely, B+ · [1] ' N+ = [B+, B+]. Hence we obtain a Lie algebrahomomorphism from g to the algebra D(N+) of differential operators on theunipotent subgroup N+ ⊂ G.

One can show that there is in fact a family of such homomorphisms param-eterized by h∗. Thus, we obtain a Lie algebra homomorphism g → Fun h∗ ⊗D(N+), and hence an algebra homomorphism U(g)→ Fun h∗⊗D(N+). Next,one shows that the image of Z(g) ⊂ U(g) lies entirely in the first factor, whichis the commutative subalgebra Fun h∗, and this is the sought-after Harish-Chandra homomorphism. One can then prove that the image is invariant

126

5.1 Overview 127

under the simple reflections si, the generators of W . After this, all thatremains to complete the proof is to give an estimate on the “size” of Z(g).

We will follow the same strategy in the affine case. Instead of an openB+-orbit of Fl, isomorphic to N+, we will consider an open orbit of the“loop space” of Fl, which is isomorphic to N+((t)). The infinitesimal ac-tion of g((t)) on N+((t)) gives rise to a homomorphism of Lie algebras g((t))→VectN+((t)). However, there is a new phenomenon which did not exist inthe finite-dimensional case: because N+((t)) is infinite-dimensional, we mustconsider a completion D(N+((t))) of the corresponding algebra of differentialoperators. It turns out that there is a cohomological obstruction to lifting ourhomomorphism to a homomorphism g((t)) → D(N+((t))). But we will show,following [FF4, F4], that this obstruction may be partially resolved, givingrise to a homomorphism of the central extension gκc of g((t)) to D(N+((t))),such that 1 7→ 1. Thus, any module over D(N+((t))) is a gκc -module of criticallevel. This gives us another explanation of the special role of the critical level.

The above homomorphism may be deformed, much like in the finite-dimen-sional case, to a homomorphism from gκc (and hence from Uκc(g)) to a (com-pleted) tensor product of D(N((t))) and Fun h∗((t)). However, we will see thatin order to make it AutO-equivariant, we need to modify the action of AutOon h∗((t)): instead of the usual action on h∗((t)) ' h∗ ⊗ ΩK we will need theaction corresponding to the ΩK-torsor Conn(Ω−ρ)D× of connections on theLH-bundle Ω−ρ on the punctured disc. Here LH is the Cartan subgroup ofthe Langlands dual group (its Lie algebra is identified with h∗). This is thefirst inkling of the appearance of connections and of the Langlands dual group.

Next, we show that under this homomorphism the image of the centerZ(g) ⊂ Uκc(g) is contained in the subalgebra FunConn(Ω−ρ)D× . Further-more, we will show that this image is equal to the centralizer of certain“screening operators” V i[1], i = 1, . . . , ` (one may think that these are theanalogues of simple reflections from the Weyl group). On the other hand,we will show that this centralizer is also equal to the algebra FunOpLG(D×),which is embedded into Fun Conn(Ω−ρ)D× via the so-called Miura transfor-mation.

Thus, we will obtain the sought-after isomorphism Z(g) ' FunOpLG(D×)of Theorem 4.3.6 identifying the center with the algebra of functions on opersfor the Langlands dual group. Actually, in what follows we will work inthe setting of vertex algebras and obtain in a similar way an isomorphismz(g) ' FunOpLG(D) of Theorem 4.3.2. (As we saw in Section 4.3.2, thisimplies the isomorphism of Theorem 4.3.6.)

The homomorphism Uκc(g)→ D(N((t)))⊗Fun h∗((t)) and its vertex algebraversion are called the free field realization of gκc

. It may also be deformedaway from the critical level, as we will see in the next chapter. This realizationwas introduced in 1986 by M. Wakimoto [W] in the case of sl2 and in 1988 byB. Feigin and the author [FF1] in the general case. It gives rise to a family

128 Free field realization

of gκ-modules, called Wakimoto modules. They have many applications inrepresentation theory, geometry and conformal field theory.

In this chapter and the next we present the construction of the free fieldrealization and the Wakimoto modules. In the case of g = sl2 the constructionis spelled out in detail in [FB], Ch. 11–12, where the reader is referred foradditional motivation and background. Here we explain it in the case of anarbitrary g. We follow the original approach of [FF4] (with some modificationsintroduced in [F4]) and prove the existence of the free field realization bycohomological methods. We note that explicit formulas for the Wakimotorealization have been given in [FF1] for g = sln and in [dBF] for general g.Another proof of the existence of the free field realization has been presentedin [FF8]. The construction has also been extended to twisted affine algebrasin [Sz].

Here is a more detailed description of the contents of this chapter. We beginin Section 5.2 with the geometric construction of representations of a simplefinite-dimensional Lie algebra g using an embedding of g into a Weyl algebrawhich is obtained from the infinitesimal action of g on the flag manifold. Thiswill serve as a prototype for the construction of Wakimoto modules presentedin the subsequent sections. In Section 5.3 we introduce the main ingredientsneeded for the constructions of Wakimoto modules: the infinite-dimensionalWeyl algebra Ag, the corresponding vertex algebra Mg and the infinitesimalaction of the loop algebra Lg on the formal loop space LU of the big cell Uof the flag manifold of g. We also introduce the local Lie algebra Ag

≤1,g ofdifferential operators on LU of order less than or equal to one. We show thatit is a non-trivial extension of the Lie algebra of local vector fields on LU bylocal functionals on LU and compute the corresponding two-cocycle (most ofthis material has already been presented in [FB], Chapter 12). In Section 5.4we give a vertex algebra interpretation of this extension. We prove that theembedding of the loop algebra Lg into the Lie algebra of local vector fields onLU may be lifted to an embedding of the central extension g to Ag

≤1,g. In or-der to do that, we need to show that the restriction of the above two-cocycleto Lg is cohomologically equivalent to the two-cocycle corresponding to itsKac–Moody central extension (of level κc). This is achieved in Section 5.5 byreplacing the standard cohomological Chevalley complex by a much smallerlocal subcomplex (where both cocycles belong). In Section 5.6 we computethe cohomology of the latter and prove that the cocycles are indeed cohomo-logically equivalent.

5.2 Finite-dimensional case

In this section we recall the realization of g-modules in the space of functionson the big cell of the flag manifold. This will serve as a blueprint for the

5.2 Finite-dimensional case 129

construction of the free field realization of the affine Kac–Moody algebras inthe following sections.

5.2.1 Flag variety

Let again g be a simple Lie algebra of rank ` with its Cartan decomposition

g = n+ ⊕ h⊕ n−, (5.2.1)

where h is the Cartan subalgebra and n± are the upper and lower nilpotentsubalgebras. Let

b± = h⊕ n±

be the upper and lower Borel subalgebras. We will follow the notation ofAppendix A.3.

Let G be the connected simply-connected Lie group corresponding to g,and N± (resp., B±) the upper and lower unipotent subgroups (resp., Borelsubgroups) of G corresponding to n± (resp., b±).

The homogeneous space Fl = G/B− is called the flag variety associatedto g. For example, for G = SLn this is the variety of full flags of subspaces ofCn: V1 ⊂ . . . ⊂ Vn−1 ⊂ Cn,dimVi = i. The group SLn acts transitively onthis variety, and the stabilizer of the flag in which Vi = span(en, . . . , en−i+1)is the subgroup B− of lower triangular matrices.

The flag variety has a unique open N+-orbit, the so-called big cell U =N+ · [1] ⊂ G/B−, which is isomorphic to N+. Since N+ is a unipotent Liegroup, the exponential map n+ → N+ is an isomorphism. Therefore N+ isisomorphic to the vector space n+. Thus, N+ is isomorphic to the affine spaceA|∆+|, where ∆+ is the set of positive roots of g. Hence the algebra FunN+

of regular functions on N+ is a free polynomial algebra. We will call a systemof coordinates yαα∈∆+ on N+ homogeneous if

h · yα = −α(h)yα, h ∈ h.

In what follows we will consider only homogeneous coordinate systems on N+.

Note that in order to define U it is sufficient to choose only a Borel subgroupB+ ofG. ThenN+ = [B+, B+] and U is the openN+-orbit in the flag manifolddefined as the variety of all Borel subgroups of G (so U is an N+-torsor). Allconstructions of this chapter make sense with the choice of B+ only, i.e.,without making the choice of an embedding of H = B+/[B+, B+] into B+

(which in particular gives the opposite Borel subgroup B−, see Section 4.2.1).However, to simplify the exposition we will fix a Cartan subgroup H ⊂ B+ aswell. We will see later on that the construction is independent of this choice.

The action of G on G/B− gives us a map from g to the Lie algebra of vector


fields on G/B−, and hence on its open dense subset U ' N+. Thus, we obtaina Lie algebra homomorphism g→ VectN+.

This homomorphism may be described explicitly as follows. Let G denotethe dense open submanifold of G consisting of elements of the form g+g−, g+ ∈N+, g− ∈ B− (note that such an expression is necessarily unique since B− ∩N+ = 1). In other words, G = p−1(U), where p is the projection G→ G/B−.Given a ∈ g, consider the one-parameter subgroup γ(ε) = exp(εa) in G. SinceG is open and dense in G, γ(ε)x ∈ G for ε in the formal neighborhood of 0,so we can write

γ(ε)−1x = Z+(ε)Z−(ε), Z+(ε) ∈ N+, Z−(ε) ∈ B−.

The factor Z+(ε) just expresses the projection of the subgroup γ(ε) ontoN+ ' U ⊂ G/B− under the map p. Then the vector field ξa (equivalently, aderivation of FunN+) corresponding to a is given by the formula

(ξaf)(x) =(d

dεf(Z+(ε))

)∣∣∣∣ε=0

. (5.2.2)

To write a formula for ξa in more concrete terms, we choose a faithful rep-resentation V of g (say, the adjoint representation). Since we only need theε-linear term in our calculation, we can and will assume that ε2 = 0. Consid-ering x ∈ N+ as a matrix whose entries are polynomials in the coordinatesyα, α ∈ ∆+, which expresses a generic element of N+ in EndV , we have

(1− εa)x = Z+(ε)Z−(ε). (5.2.3)

We find from this formula that Z+(ε) = x + εZ(1)+ , where Z(1)

+ ∈ n+, andZ−(ε) = 1 + εZ

(1)− , where Z

(1)− ∈ b−. Therefore we obtain from formulas

(5.2.2) and (5.2.3) that

ξa · x = −x(x−1ax)+, (5.2.4)

where z+ denotes the projection of an element z ∈ g onto n+ along b−.For example, let g = sl2. Then G/B− = P1, the variety of lines in C2. As

an open subset of CP1 we take

U =

C(

y

−1

)⊂ CP1

(the minus sign is chosen here for notational convenience). We obtain a Liealgebra homomorphism sl2 → VectU sending a to ξa, which can be calculatedexplicitly by the formula

(va · f)(y) =d

dεf(exp(−εa)y)|ε=0.

It is easy to find explicit formulas for the vector fields corresponding to


elements of the standard basis of sl2 (see Section 2.1.1):

e 7→ ∂

∂y, h 7→ −2y

∂

∂y, f 7→ −y2 ∂

∂y. (5.2.5)

5.2.2 The algebra of differential operators

The algebra D(U) of differential operators on U is isomorphic to the Weylalgebra with generators yα, ∂/∂yαα∈∆+ , and the standard relations[

∂

∂yα, yβ

]= δα,β ,

[∂

∂yα,∂

∂yβ

]= [yα, yβ ] = 0.

The algebra D(U) has a natural filtration D≤i(U) by the order of thedifferential operator. In particular, we have an exact sequence

0→ FunU → D≤1(U)→ VectU → 0, (5.2.6)

where FunU ' FunN+ denotes the ring of regular functions on U , and VectUdenotes the Lie algebra of vector fields on U . This sequence has a canonicalsplitting: namely, we lift ξ ∈ VectU to the unique first order differentialoperator Dξ whose symbol equals ξ and which kills the constant functions,i.e., such that Dξ · 1 = 0. Using this splitting, we obtain an embeddingg→ D≤1(N+), and hence the structure of a g-module on the space of functionsFunN+ = C[yα]α∈∆+ .

5.2.3 Verma modules and contragredient Verma modules

By construction, the action of n+ on FunN+ satisfies eα ·yα = 1 and eα ·yβ = 0unless α is less than or equal to β with respect to the usual partial orderingon the set of positive roots (for which α ≤ β if β − α is a linear combinationof positive simple roots with non-negative coefficients). Therefore, we obtainby induction that for any A ∈ FunN+ there exists P ∈ U(n+) such thatP ·A = 1.

Consider the pairing

U(n+)× FunN+ → C,

which maps (P,A) to the value of the function P · A at the identity elementof N+. This pairing is n+-invariant. Moreover, both U(n+) and FunN+ aregraded by the positive part Q+ of the root lattice of g (under the action ofthe Cartan subalgebra h):

U(n+) =⊕

γ∈Q+

U(n+)γ , FunN+ =⊕

γ∈Q+

(FunN+)−γ ,


where

Q+ =

∑i=1

niαi

∣∣∣∣∣ ni ∈ Z+

,

and this pairing is homogeneous. This means that the pairing of homogeneouselements is non-zero only if their degrees are opposite.

Let us look at the restriction of the pairing to the finite-dimensional sub-spaces U(n+)γ and (FunN+)−γ for some γ ∈ Q+. Consider he Poincare–Birkhoff–Witt basis elements

eα(1) . . . eα(k),

k∑i=1

α(i) = γ

(5.2.7)

(with respect to some lexicographic ordering) of U(n+)γ and the monomialbasis

yβ(1) . . . yβ(m),

m∑i=1

β(i) = γ

of FunN+.Using formula (5.2.4) we obtain the following formulas for the action of the

vector field corresponding to eα on FunN+:

eα 7→∂

∂yα+

∑β∈∆+,β>α

Pαβ

∂

∂yβ, (5.2.8)

where Pαβ ∈ FunN+ is a polynomial of degree α − β, which is a non-zero

element of −Q+ (this is what we mean by β > α in the above formula).Let us choose the lexicographic ordering of our monomials (5.2.7) in such

a way that α(i) > α(j) only if i > j. Then it is easy to see from formula(5.2.8) that the matrix of our pairing, restricted to degree γ, is diagonal withnon-zero entries, and so it is non-degenerate.

Therefore we find that the n+-module FunN+ is isomorphic to the restricteddual U(n+)∨ of U(n+):

U(n+)∨ def=⊕

γ∈Q+

(U(n+)γ)∗.

Now we recall the definition of the Verma modules and the contragredientVerma modules.

For each χ ∈ h∗, consider the one-dimensional representation Cχ of b± onwhich h acts according to χ, and n± acts by 0. The Verma module Mχ

with highest weight χ ∈ h∗ is the induced module

Mχ = Indgb+

Cχdef= U(g) ⊗

U(b+)Cχ.


The Cartan decomposition (5.2.1) gives us an isomorphism of vector spacesU(g) ' U(n−)⊗

CU(b+). Therefore, as an n−-module, Mχ ' U(n−).

The contragredient Verma module M∗χ with highest weight χ ∈ h∗ isdefined as the (restricted) coinduced module

M∗χ = Coindgb−

Cχdef= Homres

U(b−)(U(g),Cχ). (5.2.9)

Here, we consider homomorphisms invariant under the action of U(b−) onU(g) from the left, and the index “res” means that we consider only thoselinear maps U(g) → Cχ which are finite linear combinations of maps sup-ported on the direct summands U(b−)⊗ U(n+)γ of U(g) with respect to theisomorphism of vector spaces U(g) ' U(b−)⊗U(n+). Then, as an n+-module,M∗χ ' U(n+)∨.

5.2.4 Identification of FunN+ with M∗χ

The module M∗0 is isomorphic to FunN+ with its g-module structure definedabove. Indeed, the vector 1 ∈ FunN+ is annihilated by n+ and has weight 0with respect to h. Hence there is a non-zero homomorphism ϕ : FunN+ →M∗0sending 1 ∈ FunN+ to a non-zero vector v∗0 ∈M∗χ of weight 0. Since FunN+

is isomorphic to U(n+)∨ as an n+-module, this homomorphism is injective.Indeed, for any P ∈ FunN+ there exists u ∈ U(n+) such that U · P = 1.Therefore u · ϕ(P ) = ϕ(1) = v∗0 6= 0, and so ϕ(P ) 6= 0. But since M∗χ isalso isomorphic to U(n+)∨ as an n+-module, we find that ϕ is necessarily anisomorphism.

Now we identify the module M∗χ with an arbitrary weight χ with FunN+,where the latter is equipped with a modified action of g.

Recall that we have a canonical lifting of g to D≤1(N+), a 7→ ξa. Butthis lifting is not unique. We can modify it by adding to each ξa a functionφa ∈ FunN+ so that φa+b = φa + φb. One readily checks that the modifieddifferential operators ξa + φa satisfy the commutation relations of g if andonly if the linear map g→ FunN+ given by a 7→ φa is a one-cocycle of g withcoefficients in FunN+.

Each such lifting gives FunN+ the structure of a g-module. Let us imposethe extra condition that the modified action of h on V remains diagonalizable.This means that φh is a constant function on N+ for each h ∈ h, and thereforeour one-cocycle should be h-invariant: φ[h,a] = ξh · φa, for all h ∈ h, a ∈ g.We claim that the space of h-invariant one-cocycles of g with coefficients inC[N+] is canonically isomorphic to the first cohomology of g with coefficientsin C[N+], i.e., H1(g,C[N+]).

Indeed, it is well-known (see, e.g., [Fu]) that if a Lie subalgebra h of g actsdiagonally on g and on a g-module M , then h must act by 0 on H1(g,M).Hence H1(g,C[N+]) is equal to the quotient of the space of h-invariant one-


cocycles by its subspace of h-invariant coboundaries (i.e., those cocycles whichhave the form φb = vb · f for some f ∈ C[N+]). But it is clear that the spaceof h-invariant coboundaries is equal to 0 in our case, hence the result.

Thus, the set of liftings of g to D≤1(N+) making C[N+] into a g-modulewith diagonal action of h is naturally isomorphic to H1(g,C[N+]). Since

FunN+ = M∗0 = Coindgb−

C0,

we find from the Shapiro lemma (see [Fu], Section 5.4) that

H1(g,FunN+) ' H1(b−,C0) = (b−/[b−, b−])∗ ' h∗.

Thus, for each χ ∈ h∗ we obtain a Lie algebra homomorphism ρχ : g →D≤1(N+) and hence the structure of an h∗-graded g-module on FunN+. Letus analyze this g-module in more detail.

We have ξh · yα = −α(h)yα, α ∈ ∆+, so the weight of any monomial inFunN+ is equal to a sum of negative roots. Since our one-cocycle is h-invariant, we obtain that

ξh · φeα = φ[h,eα] = α(h)φeα , α ∈ ∆+,

so the weight of φeα has to be equal to the positive root α. Therefore φeα = 0for all α ∈ ∆+. Thus, the action of n+ on FunN+ is not modified. Onthe other hand, by construction, the action of h ∈ h is modified by h 7→h+χ(h). Therefore the vector 1 ∈ FunN+ is still annihilated by n+, but nowit has weight χ with respect to h. Hence there is a non-zero homomorphismFunN+ → M∗χ sending 1 ∈ FunN+ to a non-zero vector v∗χ ∈ M∗χ of weightχ. Since both FunN+ and M∗χ are isomorphic to U(n+)∨ as n+-modules, weobtain that this homomorphism is an isomorphism (in the same way as wedid at the beginning of this section for χ = 0). Thus, under the modifiedaction obtained via the lifting ρχ, the g-module FunN+ is isomorphic to thecontragredient Verma module M∗χ.

To summarize, we have constructed a family of Lie algebra homomorphisms

ρχ : g→ D≤1(N+)→ D(N+), χ ∈ h∗,

which give rise to homomorphisms of algebras

ρχ : U(g)→ D(N+), χ ∈ h∗.

These homomorphisms combine into a universal homomorphism of algebras

ρ : U(g)→ Fun h∗ ⊗CD(N+), (5.2.10)

such that for each χ ∈ h∗ we recover ρχ as the composition of ρ and theevaluation at χ homomorphism Fun h∗ → C along the first factor.

5.3 The case of affine algebras 135

5.2.5 Explicit formulas

Choose a basis Jaa=1,...,dim g of g. Under the homomorphism ρχ, we have

Ja 7→ Pa

(yα,

∂

∂yα

)+ fa(yα), (5.2.11)

where Pa is a polynomial in the yα’s and ∂/∂yα’s of degree one in the ∂/∂yα’s,which is independent of χ, and fa is a polynomial in the yα’s only, whichdepends on χ.

Let ei, hi, fi, i = 1, . . . , `, be the generators of g. Using formula (5.2.4) weobtain the following formulas:

ρχ(ei) =∂

∂yαi

+∑

β∈∆+

P iβ(yα)

∂

∂yβ, (5.2.12)

ρχ(hi) = −∑

β∈∆+

β(hi)yβ∂

∂yβ+ χ(hi), (5.2.13)

ρχ(fi) =∑

β∈∆+

Qiβ(yα)

∂

∂yβ+ χ(hi)yαi

, (5.2.14)

for some polynomials P iβ , Q

iβ in yα, α ∈ ∆+.

In addition, we have a Lie algebra anti-homomorphism ρR : n+ → D≤1(N+),which corresponds to the right action of n+ on N+. The differential operatorsρR(x), x ∈ n+, commute with the differential operators ρχ(x′), x′ ∈ n+ (buttheir commutation relations with ρχ(x′), x′ 6∈ n+, are complicated in general).We have

ρR(ei) =∂

∂yαi

+∑

β∈∆+

PR,iβ (yα)

∂

∂yβ

for some polynomials PR,iβ in yα, α ∈ ∆+.

5.3 The case of affine algebras

In this section we develop a similar formalism for the affine Kac–Moody al-gebras. This will enable us to construct Wakimoto modules, which is animportant step in our program.

5.3.1 The infinite-dimensional Weyl algebra

Our goal is to generalize the above construction to the case of affine Kac–Moody algebras. Let again U be the open N+-orbit of the flag manifold ofG, which we identify with the group N+ and hence with the Lie algebra n+.Consider the formal loop space LU = U((t)) as a complete topological vector


space with the basis of open neighborhoods of 0 ∈ LU formed by the subspacesU ⊗ tNC[[t]] ⊂ LU , N ∈ Z. Thus, LU is an affine ind-scheme

LU = lim−→U ⊗ tNC[[t]], N < 0.

Using the coordinates yα, α ∈ ∆+, on U , we can write

U ⊗ tNC[[t]] '

∑n≥N

yα,ntn

α∈∆+

= Spec C[yα,n]n≥N .

Therefore we obtain that the ring of functions on LU , denoted by FunLU ,is the inverse limit of the rings C[yα,n]α∈∆+,n≥N , N < 0, with respect to thenatural surjective homomorphisms

sN,M : C[yα,n]α∈∆+,n≥N → C[yα,n]α∈∆+,n≥M , N < M,

such that yα,n 7→ 0 for N ≤ n < M and yα,n 7→ yα,n, n ≥ M . This is acomplete topological ring, with the basis of open neighborhoods of 0 given bythe ideals generated by yα,n, n < N , i.e., the kernels of the homomorphismss∞,M : FunLU → FunU ⊗ tNC[[t]].

A vector field on LU is by definition a continuous linear endomorphism ξ

of FunLU which satisfies the Leibniz rule: ξ(fg) = ξ(f)g + fξ(g). In otherwords, a vector field is a linear endomorphism ξ of FunLU such that for anyM < 0 there exist N ≤M and a derivation

ξN,M : C[yα,n]α∈∆+,n≥N → C[yα,n]α∈∆+,n≥M ,

which satisfies

s∞,M (ξ · f) = ξN,M · s∞,N (f)

for all f ∈ FunLU . The space of vector fields is naturally a topological Liealgebra, which we denote by VectLU .

More concretely, an element of FunLU may be represented as a (possiblyinfinite) series ∑

m≤−M

Pα,myα,m,

where the Pα,m’s are arbitrary (finite) polynomials in yα,n, n ∈ Z. In thisformula and in analogous formulas below the summation over α ∈ ∆+ isalways understood.

The Lie algebra VectLU may also be described as follows. Identify thetangent space T0LU at the origin in LU with LU , equipped with the structureof a complete topological vector space. Then VectLU is isomorphic to thecompleted tensor product of FunLU and LU . This means that vector fieldson LU can be described more concretely as series∑

n∈ZPα,n

∂

∂yα,n,


where Pα,n ∈ FunLU satisfies the following property: for each M ≥ 0, thereexists K ≥ M such that each Pn, n ≤ −K, lies in the ideal generated by theyα,m, α ∈ ∆+,m ≤ −M .

Such an element may be written as follows:∑n≥N

Pα,n∂

∂yα,n+

∑m≤−M

yα,mVα,m, (5.3.1)

where the Pα,n’s are polynomials and the Vα,m’s are polynomial vector fields.In other words, VectLU is the completion of the Lie algebra of polynomial

vector fields in the variables yα,n, n ∈ Z, with respect to the topology inwhich the basis of open neighborhoods of 0 is formed by the subspaces IN,M

which consist of the vector fields that are linear combinations of vector fieldsPα,n∂/∂yα,n, n ≥ N , and yα,mVα,m,m ≤ −M .

The Lie bracket of vector fields is continuous with respect to this topol-ogy. This means that the commutator between two series of the form (5.3.1),computed in the standard way (term by term), is again a series of the aboveform.

5.3.2 Action of Lg on LUFrom now on we will use the notation Lg for g((t)) (L stands for “loops”).

We have a natural Lie algebra homomorphism

ρ : Lg→ VectLU ,

which may be described explicitly by the formulas that we obtained in thefinite-dimensional case, in which we replace the ordinary variables yα withthe “loop variables” yα,n. More precisely, we have the following analogue offormula (5.2.3):

(1− εA⊗ tm)x(t) = Z+(ε)Z−(ε)

where x ∈ N+((t)), Z+(ε) = x(t)+εZ(1)+ , Z(1)

+ ∈ n+((t)), and Z−(ε) = 1+εZ(1)− ,

Z(1)− ∈ b−((t)). As before, we choose a faithful finite-dimensional representa-

tion V of g and consider x(t) as a matrix whose entries are Laurent powerseries in t with coefficients in the ring of polynomials in the coordinatesyα,n, α ∈ ∆+, n ∈ Z, expressing a generic element of N+((t)) in EndV ((t)).We define ρ by the formula

ρ(a⊗ tm) · x(t) = Z(1)+ .

Then we have the following analogue of formula (5.2.4):

ρ(a⊗ tm) · x(t) = −x(t)(x(t)−1(a⊗ tm)x(t)

)+, (5.3.2)

where z+ denotes the projection of an element z ∈ g((t)) onto n+((t)) alongb−((t)).


This formula implies that for any a ∈ g the series

ρ(a(z)) =∑n∈Z

ρ(a⊗ tn)z−n−1

may be obtained from the formula for ρ0(a) = ξa by the substitution

yα 7→∑n∈Z

yα,nzn,

∂

∂yα7→∑n∈Z

∂

yα,nz−n−1.

5.3.3 The Weyl algebra

Let Ag be the Weyl algebra with generators

aα,n =∂

∂yα,n, a∗α,n = yα,−n, α ∈ ∆+, n ∈ Z,

and relations

[aα,n, a∗β,m] = δα,βδn,−m, [aα,n, aβ,m] = [a∗α,n, a

∗β,m] = 0. (5.3.3)

The change of sign of n in the definition of a∗α,n is made so as to have δn,−m

in formula (5.3.3), rather than δn,m. This will be convenient when we use thevertex algebra formalism.

Introduce the generating functions

aα(z) =∑n∈Z

aα,nz−n−1, (5.3.4)

a∗α(z) =∑n∈Z

a∗α,nz−n. (5.3.5)

Consider a topology on Ag in which the basis of open neighborhoods of 0is formed by the left ideals IN,M , N,M ∈ Z, generated by aα,n, α ∈ ∆+, n ≥N , and a∗α,m, α ∈ ∆+,m ≥ M . The completed Weyl algebra Ag is bydefinition the completion of Ag with respect to this topology. The algebra Ag

should be thought of as an analogue of the algebra of differential operatorson LU (see [FB], Chapter 12, for more details).

In concrete terms, elements of Ag may be viewed as arbitrary series of theform ∑

n≥N

Pα,naα,n +∑

m≥M

Qα,ma∗α,m, Pn, Qm ∈ Ag. (5.3.6)

Let Ag0 be the (commutative) subalgebra of Ag generated by a∗α,n, α ∈

∆+, n ∈ Z, and Ag0 its completion in Ag. Next, let Ag

≤1 be the subspace of Ag

spanned by the products of elements of Ag0 and the generators aα,n. Denote


by Ag≤1 its completion in Ag. Thus, Ag

≤1 consists of all elements P of Ag withthe property that

P mod IN,M ∈ Ag≤1 mod IN,M , ∀N,M ∈ Z.

Here is a more concrete description of Ag0 and Ag

≤1 using the realization ofAg by series of the form (5.3.6). The space Ag

0 consists of series of the form(5.3.6), where all Pα,n = 0 and Qα,m ∈ Ag

0. In other words, these are theseries which do not contain aα,n, n ∈ Z. The space Ag

≤1 consists of elementsof the form (5.3.6), where Pα,n ∈ Ag

0 and Qα,m ∈ Ag≤1.

Proposition 5.3.1 There is a short exact sequence of Lie algebras

0→ FunLU → Ag≤1 → VectLU → 0. (5.3.7)

Proof. The statement follows from the following three assertions:

(1) Ag≤1 is a Lie algebra, and Ag

0 is its ideal.

(2) Ag0 ' FunLU .

(3) There is a surjective homomorphism of Lie algebras Ag≤1 → VectLU

whose kernel is Ag0.

(1) follows from the definition of Ag≤1 and Ag

0 as completions of Ag≤1 and

Ag0, respectively, the fact that Ag

≤1 is a Lie algebra containing Ag0 as an ideal

and the continuity of the Lie bracket.(2) follows from the definitions.To prove (3), we construct a surjective linear map Ag

≤1 → VectLU whosekernel is Ag

0 and show that it is a homomorphism of Lie algebras.A general element of Ag

≤1 may be written in the form∑n≥N

Pα,naα,n +∑

m≥M1

∑k∈Km

Q(1)α,β,k,maβ,ka

∗α,m +

∑m≥M2

Q(2)α,ma

∗α,m,

for some integers N,M1,M2, where Pα,n, Q(1)α,β,k,m, Q

(2)α,m ∈ Ag

0, and eachKm ⊂ Z is a finite set. We define our map by sending this element to∑

n≥N

Pα,naα,n +∑

m≥M1

∑k∈Km

a∗α,mQ(1)α,β,k,maβ,k,

which is a well-defined element of VectLU according to its description givenat the end of Section 5.3.1.

It is clear that elements of Ag≤1 which are in the left ideal generated by

aα,n, n ≥ N , and a∗α,m, n ≥M , are mapped to the elements of VectLU whichare in the sum of the left ideal of aα,n, n ≥ N , and the right ideal of a∗α,m,m ≥M . Therefore we obtain a map of the corresponding quotients. Since elementsof these quotients are represented by finite linear combinations of elements ofAg and VectLU , respectively, we obtain that this map of quotients is a Lie


algebra homomorphism. The statement that our map is also a Lie algebrahomomorphism then follows by continuity of the Lie brackets on Ag

≤1 andVectLU with respect to the topologies defined by the respective ideals.

Thus, we obtain from Proposition 5.3.1 that Ag≤1 is an extension of the Lie

algebra VectLU by its module Ag0 = FunLU .

This extension is however different from the standard (split) extension defin-ing the Lie algebra of the usual differential operators on LU of order less thanor equal to 1 (it corresponds to another completion of differential operatorsdefined in [FB], Section 12.1.3). The reason is that the algebra Ag does notact on the space of functions on LU = U((t)) (see the discussion in [FB], Chap-ter 12, for more details). It acts instead on the module Mg defined below.This module may be thought of as the space of “delta-functions” supportedon the subspace L+U = U [[t]]. Because of that, the trick that we used inthe finite-dimensional case, of lifting a vector field to the differential operatorannihilating the constant function 1, does not work any more: there is nofunction annihilated by all the aα,n’s in the module Mg! Therefore there isno obvious splitting of the short exact sequence (5.3.7). In fact, we will seebelow that it is non-split.

Because of that we cannot expect to lift the homomorphism ρ : Lg →VectLU to a homomorphism Lg → Ag

≤1, as in the finite-dimensional case.Nevertheless, we will show that the homomorphism ρ may be lifted to a ho-momorphism

gκc → Ag≤1, 1 7→ 1,

where gκc is the affine Kac–Moody algebra at the critical level defined inSection 1.3.6.

Our immediate goal is to prove this assertion. We will begin by showing inthe next section that the image of the embedding Lg→ VectLU belongs to aLie subalgebra of “local” vector fields T g

loc ⊂ VectLU . This observation willallow us to replace the extension (5.3.7) by its “local” part, which is muchsmaller and hence more manageable.

5.4 Vertex algebra interpretation

It will be convenient to restrict ourselves to smaller, “local” subalgebras ofthe Lie algebras appearing in the short exact sequence (5.3.7). These smallersubalgebras will be sufficient for our purposes because, as we will see, theimage of g((t)) lands in the “local” version of VectLU . In order to define thesesubalgebras, we recast everything in the language of vertex algebras.

5.4 Vertex algebra interpretation 141

5.4.1 Heisenberg vertex algebra

Let Mg be the Fock representation of Ag generated by a vector |0〉 such that

aα,n|0〉 = 0, n ≥ 0; a∗α,n|0〉 = 0, n > 0.

It is clear that the action of Ag on Mg extends to a continuous action of thetopological algebra Ag (here we equip Mg with the discrete topology). More-over, Mg carries the structure of a Z+-graded vertex algebra defined as follows(see Section 2.2.2 for the definition of vertex algebras and the ReconstructionTheorem 2.2.4, which we use to prove that the structures introduced belowsatisfy the axioms of a vertex algebra):

• Z+-grading: deg aα,n = deg a∗α,n = −n,deg |0〉 = 0;• vacuum vector: |0〉;• translation operator: T |0〉 = 0, [T, aα,n] = −naα,n−1, [T, a∗α,n] =−(n− 1)a∗α,n−1;

• vertex operators:

Y (aα,−1|0〉, z) = aα(z), Y (a∗α,0|0〉, z) = a∗α(z),

Y (aα1,n1 . . . aαk,nka∗β1,m1

. . . a∗βl,ml|0〉, z) =

k∏i=1

1(−ni − 1)!

l∏j=1

1(−mj)!

·

· :∂−n1−1z aα1(z) . . . ∂

−nk−1z aαk

(z)∂−m1z a∗β1

(z) . . . ∂−mlz a∗βl

(z): .

Here we use the normal ordering operation (denoted by the columns) intro-duced in Section 2.2.5. In the general case it is defined inductively and so israther inexplicit, but in the case at hand it can be defined in a more explicitway which we now recall.

Let us call the generators aα,n, n ≥ 0, and a∗α,m,m > 0, annihilationoperators, and the generators aα,n, n < 0, and a∗α,m,m ≤ 0, creation oper-ators. A monomial P in aα,n and a∗α,n is called normally ordered if all factorsof P which are annihilation operators stand to the right of all factors of Pwhich are creation operators. Given any monomial P , we define the normallyordered monomial :P : as the monomial obtained by moving all factors of Pwhich are annihilation operators to the right, and all factors of P which arecreation operators to the left. For example,

:aα,4a∗β,−5a

∗γ,1a

∗µ,−3: = a∗β,−5a

∗µ,−3aα,4a

∗γ,1.

Note that since the annihilation operators commute with each other, it doesnot matter how we order them among themselves. The same is true forthe creation operators. This shows that :P : is well-defined by the aboveconditions.

Given two monomials P and Q, their normally ordered product is by def-inition the normally ordered monomial :PQ:. By linearity, we define the


normally ordered product of any number of vertex operators from the vertexalgebra Mg by applying the above definition to each monomial appearing ineach Fourier coefficient of the product.

Now let

U(Mg) = (Mg ⊗ C((t)))/ Im(T ⊗ 1 + Id⊗∂t).

be the Lie algebra attached to Mg as in Section 3.2.1.As explained in Section 3.2.1, U(Mg) may be viewed as the completion of

the span of all Fourier coefficients of vertex operators from Mg. Moreover, weshow in the same way as in Proposition 3.2.1 that the map

U(Mg)→ Ag, A⊗ f(z) 7→ Resz=0 Y (A, z)f(z)dz

is a homomorphism of Lie algebras.Note that U(Mg) is a Lie algebra, but not an algebra. For instance, it

contains the generators aα,n, a∗α,n of the Heisenberg algebra, but does not

contain monomials in these generators of degree greater than one. However,we will only need the Lie algebra structure on U(Mg).†

The elements of Ag0 = FunLU which lie in the image of U(Mg) are usually

called local functionals on LU . The elements of Ag which belong to U(Mg)are given by (possibly infinite) linear combinations of Fourier coefficients ofnormally ordered polynomials in aα(z), a∗α(z) and their derivatives. We referto them as local elements of Ag.

5.4.2 More canonical definition of Mg

The above definition of the vertex algebra Mg referred to a particular sys-tem of coordinates yα, α ∈ ∆+, on the group N+. If we choose a differentcoordinate system y′α, α ∈ ∆+, on N+, we obtain another Heisenberg algebrawith generators a′α,n and a∗α,n

′ and a vertex algebra M ′g. However, the vertexalgebras M ′g and Mg are canonically isomorphic to each other. In particular,it is easy to express the vertex operators a′α(z) and a∗α

′(z) in terms of aα(z)and a∗α(z). Namely, if y′α = Fα(yβ), then

a′α(z) 7→∑

γ∈∆+

:∂yγFα(a∗β(z)) aγ(z):,

a∗α′(z) 7→ Fα(a∗β(z)).

Note that the homogeneity condition on the coordinate systems severelyrestricts the possible forms of the functions Fα:

Fα = cαyα +∑

β1+...+βk=α

cβ1,...,βkyβ1 . . . yβk

,

† As in Lemma 3.2.2 it follows that eAg is isomorphic to eU(Mg), where eU is the functorintroduced in Section 3.2.3, but we will not use this fact.

5.4 Vertex algebra interpretation 143

where cβ1,...,βk∈ C and cα 6= 0.

It is also possible to define Mg without any reference to a coordinate systemon N+. Namely, we may identify Mg, as an n+((t))-module, with

Mg = Indn+((t))n+[[t]] Fun(N+[[t]]) ' U(n+ ⊗ t−1C[t−1])⊗ Fun(N+[[t]]),

where Fun(N+[[t]]) is the ring of regular functions on the pro-algebraic groupN+[[t]], considered as an n+[[t]]-module. If we choose a coordinate systemyαα∈∆+ on N+, then we obtain a coordinate system yα,nα∈∆+,n≥0 onN+[[t]]. Then u⊗P (yα,n) ∈Mg, where u ∈ U(n+⊗ t−1C[t−1]) and P (yα,n) ∈Fun(N+[[t]]) = C[yα,n]α∈∆+,n≥0, corresponds to u · P (a∗α,n) in our previousdescription of Mg.

It is straightforward to define a vertex algebra structure on Mg (see [FF8],Section 2). Namely, the vacuum vector of Mg is the vector 1⊗ 1 ∈ Mg. Thetranslation operator T is defined as the operator −∂t, which naturally actson Fun(N+[[t]]) as well as on n+((t)) preserving n+[[t]]. Next, we define thevertex operators corresponding to the elements of Mg of the form x−1|0〉,where x ∈ n+, by the formula

Y (x−1|0〉, z) =∑n∈Z

xnz−n−1,

where xn = x ⊗ tn, and we consider its action on Mg viewed as the inducedrepresentation of n+((t)). We also need to define the vertex operators

Y (P |0〉, z) =∑n∈Z

P(n)z−n−1

for P ∈ Fun(N+[[t]]). The corresponding linear operators P(n) are completelydetermined by their action on |0〉:

P(n)|0〉 = 0, n ≥ 0,

P(n)|0〉 =1

(−n− 1)!T−n−1P |0〉,

their mutual commutativity and the following commutation relations withn+((t)):

[xm, P(k)] =∑n≥0

(m

n

)(xn · P )(m+k−n).

Using the Reconstruction Theorem 2.2.4, it is easy to prove that theseformulas define a vertex algebra structure on Mg.

In fact, the same definition works if we replace N+ by any algebraic groupG. In the general case, it is natural to consider the central extension gκ of theloop algebra g((t)) corresponding to an invariant inner product κ on g definedas in Section 5.3.3. Then we have the induced module

Indbgκ

g[[t]]⊕C1 Fun(G[[t]]),


where the central element 1 acts on Fun(G[[t]]) as the identity. The corre-sponding vertex algebra is the algebra of chiral differential operators on G,considered in [GMS, AG]. As shown in [GMS, AG], in addition to the natural(left) action of gκ on this vertex algebra, there is another (right) action ofg−κ−κg , which commutes with the left action. Here κg is the Killing form ong, defined by the formula κg(x, y) = Trg(adx ad y). In the case when g = n+,there are no non-zero invariant inner products (in particular, κn+ = 0), andso we obtain a commuting right action of n+((t)) on Mg. We will use this rightaction below (see Section 6.1.1).

The above formulas in fact define a canonical vertex algebra structure on

Indn+((t))n+[[t]] Fun(U [[t]]),

which is independent of the choice of identification N+ ' U . Recall that inorder to define U we only need to fix a Borel subgroup B+ of G. Then Uis defined as the open B+-orbit of the flag manifold and so it is naturallyan N+-torsor. In order to identify U with N+ we need to choose a point inU , i.e., an opposite Borel subgroup B−, or, equivalently, a Cartan subgroupH = B+∩B− of B+. But in the above formulas we never used an identificationof N+ and U , only the canonical action of n+ on U , which determined acanonical Ln+-action on LU .

If we do not fix an identification N+ ' U , then the right action of n+ on N+

discussed above becomes an action of the “twisted” Lie algebra n+,U , wheren+,U = U ×

N+

n+. It is interesting to observe† that unlike n+, this twisted

Lie algebra n+,U has a canonical decomposition into the one-dimensional sub-spaces nα,U corresponding to the positive roots α ∈ ∆+. Indeed, for anypoint u ∈ U we have an identification U ' N+ and a Cartan subgroupHu = B+ ∩ Bu, where Bu is the stabilizer of u. The subspaces nα,U aredefined as the eigenspaces in n+ with respect to the adjoint action of Hu. Ifwe choose a different point u′ ∈ U , the identification U ' N+ and Hu willchange, but the eigenspaces will remain the same!

Therefore there exist canonical (up to a scalar) generators eRi of the right

action of nαi,U on U . We will use these operators below to define the screeningoperators, and their independence of the choice of the Cartan subgroup in B+

implies the independence of the kernel of the screening operator from anyadditional choices.

5.4.3 Local extension

For our purposes we may replace Ag, which is a very large topological algebra,by a relatively small “local part” U(Mg). Accordingly, we replace Ag

0 and Ag≤1

by their local versions Ag0,loc = Ag

0 ∩ U(Mg) and Ag≤1,loc = Ag

≤1 ∩ U(Mg).

† I thank D. Gaitsgory for pointing this out.

5.5 Computation of the two-cocycle 145

Let us describe Ag0,loc and Ag

≤1,loc more explicitly. The space Ag0,loc is

spanned (topologically) by the Fourier coefficients of all polynomials in the∂n

z a∗α(z), n ≥ 0. Note that because the a∗α,n’s commute among themselves,

these polynomials are automatically normally ordered. The space Ag≤1,loc is

spanned by the Fourier coefficients of the fields of the form

:P (a∗α(z), ∂za∗α(z), . . .)aβ(z):

(the normally ordered product of P (a∗α(z), ∂za∗α(z), . . .) and aβ(z)).

Here we use the fact that the Fourier coefficients of all fields of the form

:P (a∗α(z), ∂za∗α(z), . . .)∂m

z aβ(z): , m > 0,

may be expressed as linear combinations of the Fourier coefficients of the fieldsof the form

:P (a∗α(z), ∂za∗α(z), . . .)aβ(z): . (5.4.1)

Further, we define a local version T gloc of VectLU as the subspace which

consists of finite linear combinations of Fourier coefficients of the formal powerseries

P (a∗α(z), ∂za∗α(z), . . .)aβ(z), (5.4.2)

where aα(z) and a∗α(z) are given by formulas (5.3.4), (5.3.5).Since Ag

≤1,loc is the intersection of Lie subalgebras of Ag, it is also a Liesubalgebra of Ag. By construction, its image in VectLU under the homo-morphism Ag

≤1 → VectLU equals T gloc. Finally, the kernel of the resulting

surjective Lie algebra homomorphism Ag≤1,loc → T

gloc equals Ag

0,loc. Hence weobtain that the extension (5.3.7) restricts to the “local” extension

0→ Ag0,loc → A

g≤1,loc → T

gloc → 0. (5.4.3)

This sequence is non-split as will see below. The corresponding two-cocyclewill be computed explicitly in Lemma 5.5.2 using the Wick formula (it comesfrom the “double contractions” of the corresponding vertex operators).

According to Section 5.3.2, the image of Lg in VectLU belongs to T gloc.

We will show that the homomorphism Lg → T gloc may be lifted to a homo-

morphism gκ → Ag≤1,loc, where gκ is the central extension of Lg defined in

Section 5.3.3.

5.5 Computation of the two-cocycle

Recall that an exact sequence of Lie algebras

0→ h→ g→ g→ 0,

where h is an abelian ideal, with prescribed g-module structure, gives rise toa two-cocycle of g with coefficients in h. It is constructed as follows. Choose a


splitting ı : g→ g of this sequence (considered as a vector space), and defineσ :∧2

g→ h by the formula

σ(a, b) = ı([a, b])− [ı(a), ı(b)].

One checks that σ is a two-cocycle in the Chevalley complex of g with coef-ficients in h, and that changing the splitting ı amounts to changing σ by acoboundary.

Conversely, suppose we are given a linear functional σ :∧2

g → h. Thenwe associate to it a Lie algebra structure on the direct sum g ⊕ h. Namely,the Lie bracket of any two elements of h is equal to 0, [X,h] = X · h for allX ∈ g, h ∈ h, and

[X,Y ] = [X,Y ]g + σ(X,Y ), X, Y ∈ g.

These formulas define a Lie algebra structure on g if and only if σ is a two-cocycle in the standard Chevalley complex of g with coefficients in h. There-fore we obtain a bijection between the set of isomorphism classes of extensionsof g by h and the cohomology group H2(g, h).

5.5.1 Wick formula

Consider the extension (5.4.3). The operation of normal ordering gives us asplitting ı of this extension as vector space. Namely, ı maps the nth Fouriercoefficient of the series (5.4.2) to the nth Fourier coefficient of the series (5.4.1).To compute the corresponding two-cocycle we have to learn how to computecommutators of Fourier coefficients of generating functions of the form (5.4.1)and (5.4.2). Those may be computed from the operator product expansion(OPE) of the corresponding vertex operators. We now explain how to computethe OPEs of vertex operators using the Wick formula.

In order to state the Wick formula, we have to introduce the notion ofcontraction of two fields. In order to simplify notation, we will assume thatg = sl2 and suppress the index α in aα,n and a∗α,n. The general case is treatedin the same way.

From the commutation relations, we obtain the following OPEs:

a(z)a∗(w) =1

z − w+ :a(z)a∗(w): ,

a∗(z)a(w) = − 1z − w

+ :a∗(z)a(w): .

We view them now as identities on formal power series, in which by 1/(z−w)we understand its expansion in positive powers of w/z. Differentiating several


times, we obtain

∂nz a(z)∂

mw a∗(w) = (−1)n (n+m)!

(z − w)n+m+1+ :∂n

z a(z)∂mw a∗(w): , (5.5.1)

∂mz a∗(z)∂n

wa(w) = (−1)m+1 (n+m)!(z − w)n+m+1

+ :∂mz a∗(z)∂n

wa(w): (5.5.2)

(here again by 1/(z − w)n we understand its expansion in positive powers ofw/z).

Suppose that we are given two normally ordered monomials in a(z), a∗(z)and their derivatives. Denote them by P (z) and Q(z). A single pairingbetween P (z) and Q(w) is by definition either the pairing (∂n

z a(z), ∂mw a∗(w))

of ∂nz a(z) occurring in P (z) and ∂m

w a∗(w) occurring in Q(w), or the pairing

(∂mz a∗(z), ∂n

wa(w)) of ∂mz a∗(z) occurring in P (z) and ∂n

wa(w) occurring inQ(w). We attach to it the functions

(−1)n (n+m)!(z − w)n+m+1

and (−1)m+1 (n+m)!(z − w)n+m+1

,

respectively. A multiple pairing B is by definition a disjoint union of singlepairings. We attach to it the function fB(z, w), which is the product of thefunctions corresponding to the single pairings in B.

Note that the monomials P (z) and Q(z) may well have multiple pairings ofthe same type. For example, the monomials :a∗(z)2∂za(z): and :a(w)∂2

za∗(w):

have two different pairings of type (a∗(z), a(w)); the corresponding functionis −1/(z−w). In such a case we say that the multiplicity of this pairing is 2.Note that these two monomials also have a unique pairing (∂za(z), ∂2

wa∗(w)),

and the corresponding function is −6/(z − w)4.Given a multiple pairing B between P (z) and Q(w), we define (P (z)Q(w))B

as the product of all factors of P (z) and Q(w) which do not belong to thepairing (if there are no factors left, we set (P (z)Q(w))B = 1). The contrac-tion of P (z)Q(w) with respect to the pairing B, denoted :P (z)Q(w):B , is bydefinition the normally ordered formal power series :(P (z)Q(w))B : multipliedby the function fB(z, w). We extend this definition to the case when B is theempty set by stipulating that

:P (z)Q(w):∅ = :P (z)Q(w): .

Now we are in a position to state the Wick formula, which gives the OPEof two arbitrary normally ordered monomial vertex operators. The proof ofthis formula is straightforward and is left to the reader.

Lemma 5.5.1 Let P (z) and Q(w) be two monomials as above. Then theproduct P (z)Q(w) equals the sum of terms :P (z)Q(w):B over all pairings Bbetween P and Q including the empty one, counted with multiplicity.


Here is an example:

:a∗(z)2∂za(z): :a(w)∂2za∗(w): = :a∗(z)2∂za(z)a(w)∂2

za∗(w):−

2z − w

:a∗(z)∂za(z)∂2za∗(w):− 6

(z − w)4:a∗(z)2a(w): +

12(z − w)5

a∗(z).

5.5.2 Double contractions

Now we can compute our two-cocycle. For this, we need to apply the Wickformula to the fields of the form

:R(a∗(z), ∂za∗(z), . . .)a(z): ,

whose Fourier coefficients span the preimage of Tloc in A≤1,loc under oursplitting ı. Two fields of this form may have only single or double pairings,and therefore their OPE can be written quite explicitly.

A field of the above form may be written as Y (P (a∗n)a−1, z) (or Y (Pa−1, z)for short), where P is a polynomial in the a∗n, n ≤ 0 (recall that a∗n, n ≤ 0,corresponds to ∂−n

z a∗(z)/(−n)!).† Applying the Wick formula, we obtain

Lemma 5.5.2

Y (Pa−1, z)Y (Qa−1, w) = :Y (Pa−1, z)Y (Qa−1, w):

+∑n≥0

1(z − w)n+1

:Y (P, z)Y(

∂Q

∂a∗−n

a−1, w

):

−∑n≥0

1(z − w)n+1

:Y(

∂P

∂a∗−n

a−1, z

)Y (Q,w):

−∑

n,m≥0

1(z − w)n+m+2

Y

(∂P

∂a∗−n

, z

)Y

(∂Q

∂a∗−m

, w

).

Note that we do not need to put normal ordering in the last summand.Using this formula and the commutation relations (3.2.2), we can now easily

obtain the commutators of the Fourier coefficients of the fields Y (Pa−1, z) andY (Qa−1, w).

The first two terms in the right hand side of the formula in Lemma 5.5.2correspond to single contractions between Y (Pa−1, z) and Y (Qa−1, w). Thepart in the commutator of the Fourier coefficients induced by these terms willbe exactly the same as the commutator of the corresponding vector fields,computed in Tloc. Thus, we see that the discrepancy between the commu-tators in A≤1,loc and in Tloc (as measured by our two-cocycle) is due to the

† In what follows we will write, by abuse of notation, Y (A, z) for the series that, strictlyspeaking, should be denoted by Y [A, z]. The reason is that the homomorphism U(Mg) →End Mg, A[n] 7→ A(n) is injective, and so we do not lose anything by considering the image

of U(Mg) in End Mg.


last term in the formula from Lemma 5.5.2, which comes from the doublecontractions between Y (Pa−1, z) and Y (Qa−1, w).

Explicitly, we obtain the following formula for our two-cocycle

ω((Pa−1)[k], (Qa−1)[s]) = (5.5.3)

−∑

n,m≥0

Resw=01

(n+m+ 1)!∂n+m+1

z Y

(∂P

∂a∗−n

, z

)Y

(∂Q

∂a∗−m

, w

)zkws

∣∣∣∣z=w

dw.

5.5.3 The extension is non-split

For example, let us compute the cocycle for the elements hn defined by theformula

Y (a∗0a−1, z) = :a∗(z)a(z): =∑n∈Z

hnz−n−1,

so that

hn =∑k∈Z

:a∗−kan+k: .

According to formula (5.5.3), we have

ω(hn,hm) = −Resw=0 nwn+m−1dw = −nδn,−m.

As the single contraction terms cancel each other, we obtain the followingcommutation relations in A≤1,loc:

[hn,hm] = −nδn,−m.

On the other hand, we find that the image of hn in Tloc is the vector field

hn =∑k∈Z

yk∂

∂yn+k.

These vector fields commute with each other.If the sequence (5.3.7) (or (5.4.3)) were split, then we would be able to

find “correction terms” fm ∈ Fun C((t)) (or in A0,loc) such that the liftinghn 7→ hn + fn preserves the Lie brackets, that is,

[hn + fn,hm + fm] = 0, ∀n,m ∈ Z.

But this is equivalent to the formula

hn · fm − hm · fn = −nδn,−m.

But since hn is a linear vector field (i.e., linear in the coordinates yk on LU),the left hand side of this formula cannot be a non-zero constant for any choiceof fm, n ∈ Z.

Thus, we find that the sequences (5.3.7) and (5.4.3) do not split as exactsequences of Lie algebras.


5.5.4 A reminder on cohomology

Thus, we cannot expect to be able to lift the homomorphism ρ : Lg → Tloc

to a homomorphism Lg → Ag≤1,loc, as in the finite-dimensional case. The

next best thing we can hope to accomplish is to lift it to a homomorphismfrom the central extension gκ of Lg to Ag

≤1,loc. Let us see what cohomologicalcondition corresponds to the existence of such a lifting.

So let again

0→ h→ l→ l→ 0

be an extension of Lie algebras, where h is an abelian Lie subalgebra and anideal in l. Choosing a splitting ı of this sequence considered as a vector spacewe define a two-cocycle of l with coefficients in h as in Section 5.5. Supposethat we are given a Lie algebra homomorphism α : g→ l for some Lie algebrag. Pulling back our two-cocycle under α we obtain a two-cocycle of g withcoefficients in h.

On the other hand, given a homomorphism h′i−→ h of g-modules we obtain

a map i∗ between the spaces of two-cocycles of g with coefficients in h and h′.The corresponding map of the cohomology groups H2(g, h′) → H2(g, h) willalso be denoted by i∗.

Lemma 5.5.3 Suppose that we are given a two-cocycle σ of g with coefficientsin h′ such that the cohomology classes of i∗(σ) and ω are equal in H2(g, h′).Denote by g the extension of g by h′ corresponding to σ defined as above. Thenthe map g→ l may be augmented to a map of commutative diagrams

0 −−−−→ h −−−−→ l −−−−→ l −−−−→ 0x x x0 −−−−→ h′ −−−−→ g −−−−→ g −−−−→ 0.

(5.5.4)

Moreover, the set of isomorphism classes of such diagrams is a torsor overH1(g, h).

Proof. If the cohomology classes of i∗(σ) and ω coincide, then i∗(σ)+dγ =ω, where γ is a one-cochain, i.e., a linear functional g → h. Define a linearmap β : g → l as follows. By definition, we have a splitting g = g ⊕ h′ as avector space. We set β(X) = ı(α(X))+γ(X) for all X ∈ g and β(h) = i(h) forall h ∈ h. Then the above equality of cocycles implies that β is a Lie algebrahomomorphism which makes the diagram (5.5.4) commutative. However, thechoice of γ is not unique as we may modify it by adding to it an arbitraryone-cocycle γ′. But the homomorphisms corresponding to γ and to γ + γ′,where γ′ is a coboundary, lead to isomorphic diagrams. This implies that theset of isomorphism classes of such diagrams is a torsor over H1(g, h).

5.6 Comparison of cohomology classes 151

5.5.5 Two cocycles

Restricting the two-cocycle ω of Tloc with coefficients in Ag0,loc corresponding

to the extension (5.4.3) to Lg ⊂ Tloc, we obtain a two-cocycle of Lg withcoefficients in Ag

0,loc. We also denote it by ω. The Lg-module Ag0,loc contains

the trivial subrepresentation C, i.e., the span of |0〉[−1] (which we view as

the constant function on LU), and the inclusion C i−→ Ag0,loc induces a map

i∗ of the corresponding spaces of two-cocycles and the cohomology groupsH2(Lg,C)→ H2(Lg,Ag

0,loc).As we discussed in Section 5.3.3, the cohomology group H2(Lg,C) is one-

dimensional and is isomorphic to the space of invariant inner products on g.We denote by σ the class corresponding to the inner product κc = − 1

2κg,where κg denotes the Killing form on g. Thus, by definition,

κc(x, y) = −12

Tr(adx ad y).

The cocycle σ is then given by the formula

σ(Jan , J

bm) = nδn,−mκc(Ja, Jb) (5.5.5)

(see Section 5.3.3).In the next section we will show that the cohomology classes of i∗(σ) and

ω are equal. Lemma 5.5.3 will then imply that there exists a family of Liealgebra homomorphisms gκc → A

g≤1,loc such that 1 7→ 1.

5.6 Comparison of cohomology classes

Unfortunately, the Chevalley complex that calculates H2(Lg,Ag0,loc) is un-

manageably large. So as the first step we will show in the next section thatω and σ actually both belong to a much smaller subcomplex, where they aremore easily compared.

5.6.1 Clifford algebras

Choose a basis Jaa=1,...,dim g of g, and set Jan = Ja ⊗ tn. Introduce the

Clifford algebra with generators ψa,n, ψ∗a,m, a = 1, . . . ,dim g;m,n ∈ Z, with

anti-commutation relations

[ψa,n, ψb,n]+ = [ψ∗a,n, ψ∗b,m]+ = 0, [ψa,n, ψ

∗b,m]+ = δa,bδn,−m.

Let∧

g be the module over this Clifford algebra generated by a vector |0〉such that

ψa,n|0〉 = 0, n > 0, ψ∗a,n|0〉 = 0, n ≥ 0.

Then∧

g carries the following structure of a Z+-graded vertex superalgebra(see [FB], Section 15.1.1):


• Z+-grading: degψa,n = degψ∗a,n = −n,deg |0〉 = 0;• vacuum vector: |0〉;• translation operator: T |0〉 = 0, [T, ψa,n] = −nψa,n−1, [T, ψ∗a,n] =−(n− 1)ψ∗a,n−1;

• vertex operators:

Y (ψa,−1|0〉, z) = ψa(z) =∑n∈Z

ψa,nz−n−1,

Y (ψ∗a,0|0〉, z) = ψ∗a(z) =∑n∈Z

ψ∗a,nz−n,

Y (ψa1,n1 . . . ψak,nkψ∗b1,m1

. . . ψ∗bl,ml|0〉, z) =

k∏i=1

1(−ni − 1)!

l∏j=1

1(−mj)!

·

:∂−n1−1z ψa1(z) . . . ∂

−nk−1z ψak

(z)∂−m1z ψ∗b1(z) . . . ∂

−mlz ψ∗bl

(z): .

The tensor product of two vertex superalgebras is naturally a vertex super-algebra (see Lemma 1.3.6 of [FB]), and so Mg ⊗

∧g is a vertex superalgebra.

5.6.2 The local Chevalley complex

The ordinary Chevalley complex computing the cohomology H•(Lg,Ag0,loc) is

C•(Lg,Ag0,loc) =

⊕i≥0 C

i(Lg,Ag0,loc), where

Ci(Lg,Ag0,loc) = Homcont(

∧i(Lg),Ag

0,loc),

where∧i(Lg) stands for the natural completion of the ordinary ith exterior

power of the topological vector space Lg, and we consider all continuous linearmaps. For f ∈ Ag

0,loc we denote by ψ∗b1,−k1. . . ψ∗bi,−ki

f the linear functionalφ ∈ Homcont(

∧i(Lg),Ag0,loc) defined by the formula

φ(Ja1m1∧ . . . ∧ Jai

mi) =

(−1)l(τ)f, ((a1,m1), . . . , (ai,mi)) = τ((b1, k1), . . . , (bi, ki)),

0, ((a1,m1), . . . , (ai,mi)) 6= τ((b1, k1), . . . , (bi, ki)),(5.6.1)

where τ runs over the symmetric group on i letters and l(τ) is the length of τ .Then any element of the space Ci(Lg,Ag

0,loc) may be written as a (possiblyinfinite) linear combination of terms of this form.

The differential d : Ci(Lg,Ag0,loc) → Ci+1(Lg,Ag

0,loc) is given by the for-mula

(dφ)(X1, . . . , Xi+1) =i+1∑j=1

(−1)j+1Xjφ(X1, . . . , Xj , . . . , Xi+1)

+∑j<k

(−1)j+k+1φ([Xj , Xk], X1, . . . , Xj , . . . , Xk, . . . , Xi+1).


It follows from the definition of the vertex operators that the linear maps∫Y (ψ∗a1,n1

· · ·ψ∗ai,nia∗α1,m1

· · · a∗αj ,mj|0〉, z) dz, np ≤ 0,mp ≤ 0. (5.6.2)

from∧i(Lg) to Ag

0,loc are continuous. Here and below we will use the notation∫f(z)dz = Resz=0 f(z)dz.

Let Ciloc(Lg,Ag

0,loc) be the subspace of the space Homcont(∧i(Lg),Ag

0,loc),spanned by all linear maps of the form (5.6.2).

Lemma 5.6.1 The Chevalley differential maps the subspace Ciloc(Lg,Ag

0,loc) ⊂Ci(Lg,Ag

0,loc) to Ci+1loc (Lg,Ag

0,loc) ⊂ Ci+1(Lg,Ag0,loc), and so

C•loc(Lg,Ag0,loc) =

⊕i≥0

Ciloc(Lg,Ag

0,loc)

is a subcomplex of C•(Lg,Ag0,loc).

Proof. The action of the differential d on∫Y (A, z)dz, where A is of the

form (5.6.2), may be written as the commutator[∫Q(z)dz,

∫Y (A, z)dz

],

where Q(z) is the field

Q(z) =∑

a

Ja(z)ψ∗a(z)− 12

∑a,b,c

µabc :ψ∗a(z)ψ∗b (z)ψc(z): , (5.6.3)

where µabc denotes the structure constants of g. Therefore it is equal to

Y(∫Q(z)dz ·A, z

), which is of the form (5.6.2).

We call Ciloc(Lg,Ag

0,loc) the local subcomplex of C•(Lg,Ag0,loc).

Lemma 5.6.2 Both ω and i∗(σ) belong to C2loc(Lg,Ag

0,loc).

Proof. We begin with σ, which is given by formula (5.5.5). Therefore thecocycle i∗(σ) is equal to the following element of C2

loc(Lg,Ag0,loc):

i∗(σ) =∑a≤b

∑n∈Z

κc(Ja, Jb)nψ∗a,−nψ∗b,n

=∑a≤b

κc(Ja, Jb)∫Y (ψ∗a,−1ψ

∗b,0|0〉, z) dz. (5.6.4)

Next we consider ω. Combining the discussion of Section 5.3.2 with formulasof Section 5.2.5, we obtain that

ı(Jan) =

∑β∈∆+

∫Y (Rβ

a(a∗α,0)aβ,−1|0〉, z) zndz,


where Rβa is a polynomial. Then formula (5.5.3) implies that

ω(Jan , J

bm) = −

∑α,β∈∆+

∫ (Y

(T∂Rα

a

∂a∗β,0

·∂Rβ

b

∂a∗α,0

, w

)wn+m

+ nY

(∂Rα

a

∂a∗β,0

∂Rβb

∂a∗α,0

, w

)wn+m−1

)dw.

Therefore

ω = −∑

a≤b;α,β∈∆+

∫ (Y

(ψ∗a,0ψ

∗b,0T

∂Rαa

∂a∗β,0

·∂Rβ

b

∂a∗α,0

, z

)(5.6.5)

+ Y

(ψ∗a,−1ψ

∗b,0

∂Rαa

∂a∗β,0

∂Rβb

∂a∗α,0

, z

))dz

(in the last two formulas we have omitted |0〉). Hence it belongs to the spaceC2

loc(Lg,Ag0,loc).

We need to show that the cocycles i∗(σ) and ω represent the same coho-mology class in the local complex C•loc(Lg,Ag

0,loc). We will show that this isequivalent to checking that the restrictions of these cocycles to the Lie sub-algebra Lh ⊂ Lg are the same. These restrictions can be easily computedand we indeed obtain that they coincide. The passage to Lh is achieved by aversion of the Shapiro lemma, as we explain in the next section.

5.6.3 Another complex

Given a Lie algebra l, we denote the Lie subalgebra l[[t]] of Ll = l((t)) by L+l.Observe that the Lie algebra L+g acts naturally on the space

Mg,+ = C[a∗α,n]α∈∆+,n≤0 ' C[yα,n]α∈∆+,n≥0,

which is identified with the ring of functions on the space U [[t]] ' N+[[t]]. Weidentify the standard Chevalley complex

C•(L+g,Mg,+) = Homcont(∧•L+g,Mg,+)

with the tensor product Mg,+ ⊗∧•

g,+, where∧•g,+ =

∧(ψ∗a,n)n≤0.

Introduce a superderivation T on C•(L+g,Mg,+) acting by the formulas

T · a∗a,n = −(n− 1)a∗a,n−1, T · ψ∗a,n = −(n− 1)ψ∗a,n−1.

We have a linear map∫: C•(L+g,Mg,+)→ C•loc(Lg,Ag

0,loc)


sending A ∈ C•(L+g,Mg,+) to∫Y (A, z)dz (recall that

∫picks out the (−1)st

Fourier coefficient of a formal power series).Recall that in any vertex algebra V we have the identity Y (TA, z) =

∂zY (A, z). Hence if A ∈ ImT , then∫Y (A, z)dz = 0.

Lemma 5.6.3 The map∫

defines an isomorphism

C•loc(Lg,Ag0,loc) ' C

•(L+g,Mg,+)/(ImT + C),

and KerT = C. Moreover, the following diagram is commutative:

C•(L+g,Mg,+) T−−−−→ C•(L+g,Mg,+)R

−−−−→ C•loc(Lg,Ag0,loc)

d

x d

x d

xC•(L+g,Mg,+) T−−−−→ C•(L+g,Mg,+)

R−−−−→ C•loc(Lg,Ag

0,loc).

Proof. It is easy to see that the differential of the Chevalley complexC•(L+g,Mg,+) acts by the formula A 7→

∫Q(z)dz ·A, where Q(z) is given by

formula (5.6.3). The lemma now follows from the argument used in the proofof Lemma 5.6.1.

Consider the double complex

C −−−−→ C•(L+g,Mg,+) T−−−−→ C•(L+g,Mg,+) −−−−→ C

d

x d

xC −−−−→ C•(L+g,Mg,+) T−−−−→ C•(L+g,Mg,+) −−−−→ C.

According to the lemma, the cohomology of the complex C•loc(Lg,Ag0,loc) is

given by the second term of the spectral sequence, in which the zeroth differ-ential is vertical.

We start by computing the first term of this spectral sequence. Let usobserve that the Lie algebra L+n+ has a Q × Z+ grading, where Q is theroot lattice of g, defined by the formulas deg eα,n = (α, n). Therefore theuniversal enveloping algebra U(L+n+) is also Z+ × Q-graded. Moreover, allhomogeneous components U(L+n+)(γ,n), for γ ∈ Q,n ∈ Z+, of U(L+n+) arefinite-dimensional. The corresponding restricted dual space is

U(L+n+)∨ =⊕

(γ,n)∈Q×Z+

(U(L+n+)(γ,n))∗.

It carries a natural structure of L+n+-module.Now let

CoindL+gL+b−

C def= HomresU(L+b−)(U(L+g),C)

(compare with formula (5.2.9)). The right hand side consists of all “restricted”linear functionals U(L+g) → C invariant under the action of U(b−) on U(g)


from the left, where “restricted” means that we consider only those func-tionals which are finite linear combinations of maps supported on the directsummands U(b−) ⊗ U(n+)(γ,n) of U(g), with respect to the isomorphism ofvector spaces U(g) ' U(b−) ⊗ U(n+). In other words, as an L+n+-module,CoindL+g

L+b−C ' U(L+n+)∨.

Lemma 5.6.4 The L+g-module Mg,+ is isomorphic to the coinduced moduleCoindL+g

L+b−C.

Proof. We apply verbatim the argument used in our proof given in Sec-tions 5.2.3 and 5.2.4 that FunN+ is isomorphic to Coindg

b−C as a g-modules.

First we prove that Mg,+ = FunL+N+ is isomorphic to U(L+n+)∨ as anL+n+-module by showing that the monomial basis in Mg,+ is dual to a PBWmonomial basis in U(L+n+), with respect to an L+n+-invariant pairing of thetwo modules. We then use this fact to prove that the natural homomorphismMg,+ → CoindL+g

L+b−C is an isomorphism.

Define a map of complexes

µ′ : C•(L+g,Mg,+)→ C•(L+b−,C)

as follows. If γ is an i-cochain in the complex

C•(L+g,Mg,+) = Homcont(∧

iL+g,Mg,+),

then µ′(γ) is by definition the restriction of γ to∧i

L+b− composed withthe natural projection Mg,+ = CoindL+g

L+b−C → C. It is clear that µ′ is a

morphism of complexes. The following is an example of the Shapiro lemma(see [Fu], Section 5.4, for the proof).

Lemma 5.6.5 The map µ′ induces an isomorphism at the level of cohomolo-gies, i.e.,

H•(L+g,Mg,+) ' H•(L+g,CoindL+gL+b−

C) ' H•(L+b−,C). (5.6.6)

Now we compute the right hand side of (5.6.6). Since L+n− ⊂ L+b− is anideal, and L+b−/L+n− ' L+h, we obtain from the Serre–Hochschild spectralsequence (see [Fu], Section 5.1) that

Hn(L+b−,C) =⊕

p+q=n

Hp(L+h,Hq(L+n−,C)).

But h⊗ 1 ∈ L+h acts diagonally on H•(L+n−,C), inducing an inner grading.According to [Fu], Section 5.2,

Hp(L+h,Hq(L+n−,C)) = Hp(L+h,Hq(L+n−,C)0),

where Hq(L+n−,C)0 is the subspace where h ⊗ 1 acts by 0. Clearly, thespace H0(L+n−,C)0 is the one-dimensional subspace of the scalars C ⊂


H0(L+n−,C), and for q 6= 0 we have Hq(L+n−,C)0 = 0. Thus, we findthat

Hp(L+h,Hq(L+n−,C)) = Hp(L+h,C).

Furthermore, we have the following result. Define a map of complexes

µ : C•(L+g,Mg,+)→ C•(L+h,C) (5.6.7)

as follows. If γ is an i-cochain in the complex

C•(L+g,Mg,+) = Homcont(∧

iL+g,Mg,+),

then µ(γ) is by definition the restriction of γ to∧i

L+h composed with thenatural projection

p : Mg,+ = CoindL+gL+b−

C→ C. (5.6.8)

Lemma 5.6.6 The map µ induces an isomorphism at the level of cohomolo-gies, i.e.,

H•(L+g,Mg,+) ' H•(L+h,C).

In particular, the cohomology class of a cocycle in the complex Ci(L+g,Mg,+)is uniquely determined by its restriction to

∧iL+h.

Proof. It follows from the construction of the Serre–Hochschild spectralsequence (see [Fu], Section 5.1) that the restriction map C•(L+b−,C) →C•(L+h,C) induces an isomorphism at the level of cohomologies. The state-ment of the lemma now follows by combining this with Lemma 5.6.5.

Since L+h is abelian, we have

H•(L+h,C) =∧•(L+h)∗,

and so

H•(L+g,Mg,+) '∧•(L+h)∗.

It is clear that this isomorphism is compatible with the action of T on bothsides. The kernel of T acting on the right hand side is equal to the subspaceof scalars. Therefore we obtain

H•loc(Lg,Ag0,loc) '

∧•(L+h)/(ImT + C).

5.6.4 Restricting the cocycles

Any cocycle in Ciloc(Lg,Ag

0,loc) is a cocycle in


∧i(Lg),Ag

0,loc),

and as such it may be restricted to∧

i(Lh).


The following lemma is the crucial result, which will enable us to show thatω and i∗(σ) define the same cohomology class.

Lemma 5.6.7 Any two cocycles in Ciloc(Lg,Ag

0,loc), whose restrictions to∧i(Lh) coincide, represent the same cohomology class.

Proof. We need to show that any cocycle φ in Ciloc(Lg,Ag

0,loc), whoserestriction to

∧i(Lh) is equal to zero, is equivalent to the zero cocycle. Ac-

cording to the above computation, φ may be written as∫Y (A, z)dz, where

A is a cocycle in Ci(L+g,Mg,+) = Homcont(∧i

L+g,Mg,+). But then the re-striction of A to

∧iL+h ⊂

∧iL+g, denoted by A, must be in the image of the

operator T , and hence so is p(A) = µ(A) ∈∧i(L+h)∗ (here p is the projection

defined in (5.6.8) and µ is the map defined in (5.6.7)). Thus, µ(A) = T (h) forsome h ∈

∧i(L+h)∗. Since T commutes with the differential and the kernelof T on

∧i(L+h)∗ consists of the scalars, we obtain that h is also a cocycle inCi(L+h,C) =

∧i(L+h)∗.According to Lemma 5.6.6, the map µ induces an isomorphism on the coho-

mologies. Hence h is equal to µ(B) for some cocycle B in Ci(L+g,Mg,+). Itis clear from the definition that µ commutes with the action of the translationoperator T . Therefore it follows that the cocycles A and T (B) are equivalentin Ci(L+g,Mg,+). But then φ is equivalent to the zero cocycle.

Now we are ready to prove the main result of this chapter.

Theorem 5.6.8 The cocycles ω and i∗(σ) represent the same cohomologyclass in H2

loc(Lg,Ag0,loc). Therefore there exists a lifting of the homomorphism

Lg → T gloc to a homomorphism gκc → A

g≤1,loc such that 1 7→ 1. Moreover,

this homomorphism may be chosen in such a way that

Ja(z) 7→ Y (Pa(a∗α,0, aβ,−1)|0〉, z) + Y (Ba, z), (5.6.9)

where Pa is the polynomial introduced in formula (5.2.11) and Ba is a polyno-mial in a∗α,n, n ≤ 0, of degree 1 (with respect to the assignment deg a∗α,n = −n).

Proof. By Lemma 5.6.7, it suffices to check that the restrictions of ω andi∗(σ) to

∧2(Lh) coincide. We have

i∗(σ)(hn, h′m) = nκc(h, h′)δn,−m

for all h, h′ ∈ h. Now let us compute the restriction of ω. We find from theformulas given in Section 5.2.5 and Section 5.3.2 that

ı(h(z)) = −∑

α∈∆+

α(h) :a∗α(z)aα(z): (5.6.10)

(recall that h(z) =∑

n∈Z hnz−n−1). Therefore we find, in the same way as in


the computation at the end of Section 5.5.3, that

ω(hn, h′m) = −nδn,−m

∑α∈∆+

α(h)α(h′).

Now the key observation is that∑α∈∆+

α(h)α(h′) = κc(h, h′), (5.6.11)

because by definition κc(·, ·) = − 12κg(·, ·) and for the Killing form κg we have

κg(h, h′) = 2∑

α∈∆+

α(h)α(h′).

Therefore we find that

ω(hn, h′m) = nκc(h, h′)δn,−m,

and so the restrictions of the cocycles ω and i∗(σ) to∧2(Lh) coincide. By

Lemma 5.6.7, they represent the same class in H2loc(Lg,Ag

0,loc).Hence there exists γ ∈ C1

loc(Lg,Ag0,loc) such that ω + dγ = i∗(σ). We may

write γ as

γ =∑

a

∫ψ∗a(z)Y (Ba, z) dz, Ba ∈Mg,+.

It follows from the computations made in the proof of Lemma 5.6.2 thatboth ω and i∗(σ) are homogeneous elements of C2

loc(Lg,Ag0,loc) of degree 0.

Therefore γ may be chosen to be of degree 0, i.e., Ba may be chosen to be ofdegree 1.

Then by Lemma 5.5.3 formulas (5.6.9) define a homomorphism of Lie alge-bras gκc → A

g≤1,loc of the form (5.6.9) such that 1 7→ 1. This completes the

proof.

5.6.5 Obstruction for other flag varieties

The key fact that we used in the proof above is that the inner product on h

defined by the left hand side of formula (5.6.11) is the restriction to h of aninvariant inner product on g. We then find that this invariant inner productis the critical inner product κc.

Let us observe that the necessary and sufficient condition for an inner prod-uct on h to be equal to the restriction to h of an invariant inner product on g

is that it is W -invariant, where W is the Weyl group of g acting on h. In thecase of the inner product defined by the left hand side of formula (5.6.11) thisis obvious as the set ∆+ is W -invariant. But this observation also singles outthe “full” flag variety G/B− as the only possible flag variety of G for whichthe cohomological obstruction may be overcome.


Indeed, consider a more general flag variety G/P , where P is a parabolicsubgroup. For example, in the case of g = sln conjugacy classes of parabolicsubgroups correspond to non-trivial partitions of n. The standard (lower)parabolic subgroup corresponding to a partition n = n1 + . . . + nk consistsof “block lower triangular” matrices with the blocks of sizes n1, . . . , nk. Thecorresponding flag variety is the variety of flags V1 ⊂ . . . ⊂ Vk−1 ⊂ Cn, wheredimVi = n1 + . . .+ ni.

Let P be a parabolic subgroup that is not a Borel subgroup and p thecorresponding Lie subalgebra of g. We may assume without loss of generalitythat p contains b− and is invariant under the adjoint action of H. Then wehave a decomposition g = p⊕n, where n is a Lie subalgebra of n+, spanned bygenerators eα, where α runs over a proper subset ∆P ⊂ ∆+. We can developthe same formalism for G/P as for G/B−. The Lie subalgebra n will now playthe role of n+: its Lie group is isomorphic to an open dense subset of G/P , etc.In particular, formula (5.6.10) is modified: instead of the summation over ∆+

we have the summation only over ∆S . Therefore in the corresponding cocyclewe obtain the inner product on h given by the formula

νP (h, h′) =∑

α∈∆P

α(h)α(h′).

But no proper subset of ∆+ is W -invariant. Therefore the inner product νP

is not W -invariant and hence cannot be obtained as the restriction to h ofan invariant inner product on g. Therefore our argument breaks down, andwe see that in the case of a parabolic subgroup P other than the Borel wecannot lift the homomorphism from Lg to the corresponding Lie algebra ofvector fields to a homomorphism from a central extension gκ to the algebraof differential operators.

This illustrates the special role played by the full flag variety G/B−. It isclosely related to the fact that the first Pontryagin class of G/B− vanisheswhile it is non-zero for other flag varieties, which leads to an “anomaly.”Nevertheless, we will show in Section 6.3 how to overcome this anomaly inthe general case. The basic idea is to add additional representations of theaffinization of the Levi subgroup m of p. By judiciously choosing the levels ofthese representations, we cancel the anomaly.

To summarize the results of this chapter, we have now obtained a “free fieldrealization” homomorphism from gκc

to the Lie algebra Ag≤1,loc of Fourier

coefficients of vertex operators built from “free fields” aα(z) and a∗α(z) cor-responding to differential operators on the flag variety G/B−. In the nextchapter we will interpret this homomorphism in the vertex algebra language.

6

Wakimoto modules

In this chapter we extend the free field realization constructed above to non-critical levels and recast it in the vertex algebra language. In these termsthe free field realization amounts to a homomorphism from the vertex algebraVκ(g) associated to gκ to a “free field” vertex algebra Mg ⊗ πκ−κc

0 associatedto an infinite-dimensional Heisenberg Lie algebra.

Modules over the vertex algebraMg⊗πκ−κc0 now become gκ-modules, which

we call the Wakimoto modules. They were first constructed by M. Wakimoto[W] in the case g = sl2 and by B. Feigin and the author [FF1, FF4] for generalg. The Wakimoto modules may be viewed as representations of g which are“semi-infinitely induced” from representations of its Heisenberg subalgebra h.In contrast to the usual induction, however, the level of the h-module getsshifted by the critical value κc. In particular, if we start with an h-moduleof level zero (e.g., a one-dimensional module corresponding to a character ofthe abelian Lie algebra Lh), then the resulting g-module will be at the criticallevel. This is the reason why it is the critical level, and not the zero level, asone might naively expect, that is the “middle point” among all levels.

The Wakimoto modules (of critical and non-critical levels) will be crucialin achieving our ultimate goal: describing the center of Vκc

(g) (see Chapter8).

Here is a more detailed description of the material of this chapter. Westart in Section 6.1 with the case of critical level. Our main result is theexistence of a vertex algebra homomorphism Vκc(g) → Mg ⊗ π0, where π0 isthe commutative vertex algebra associated to Lh. Using this homomorphism,we construct a g-module structure of critical level on the tensor product Mg⊗N , where N is an arbitrary (smooth) Lh-module. These are the Wakimotomodules of critical level (we will see below that they are irreducible for genericvalues of the parameters).

In Section 6.2 we generalize this construction to arbitrary levels. We provethe existence of a homomorphism of vertex algebras

wκ : Vκ(g)→Mg ⊗ πκ−κc0 ,

161

162 Wakimoto modules

where πκ−κc0 is the vertex algebra associated to the Heisenberg Lie algebra

hκ−κc. As in our earlier discussion of the center, we pay special attention to the

action of the group AutO of coordinate changes on the Wakimoto modules.This is necessary to achieve a coordinate-independent construction of thesemodules, which we will need to obtain a coordinate-independent description ofthe center of Vκc

(g). To do this, we describe the (quasi)conformal structure onMg⊗πκ−κc

0 corresponding to the Segal–Sugawara (quasi)conformal structureon Vκ(g).

Next, we extend the construction of the Wakimoto modules in Section 6.3 toa more general context in which the Lie subalgebra h is replaced by a centralextension of the loop algebra of the Levi subalgebra of an arbitrary parabolicLie subalgebra of g following the ideas of [FF4]. Thus, we establish a “semi-infinite parabolic induction” pattern for representations of affine Kac–Moodyalgebras, similar to the parabolic induction for reductive groups over localnon-archimedian fields.

6.1 Wakimoto modules of critical level

In this section we will recast the free field realization obtained in the previouschapter as a homomorphism of vertex algebras Vκc

(g) → Mg ⊗ π0, where π0

is the commutative vertex algebra associated to Lh.

6.1.1 Homomorphism of vertex algebras

Recall that to the affine Kac–Moody algebra gκ we associate the vacuum gκ-module Vκ(g), as defined in Section 2.2.4. We have shown in Theorem 2.2.2that Vκ(g) is a Z-graded vertex algebra.

We wish to interpret a Lie algebra homomorphism gκc → Ag≤1,loc in terms

of a homomorphism of vertex algebras.

Lemma 6.1.1 Defining a homomorphism of Z-graded vertex algebras Vκ(g)→V is equivalent to choosing vectors Ja

−1|0〉V ∈ V, a = 1, . . . ,dim g, of degree 1such that the Fourier coefficients Ja

n of the vertex operators

Y (Ja−1|0〉V , z) =

∑n∈Z

Janz−n−1,

satisfy the relations (1.3.4) with 1 = 1.

Proof. Given a homomorphism ρ : Vκ(g) → V , set Ja−1|0〉V = ρ(Ja

−1|0〉).The fact that ρ is a homomorphism of vertex algebras implies that the OPEsof the vertex operators Y (Ja

−1|0〉V , z) will be the same as those of the ver-tex operators Ja(z), and hence the commutation relations of their Fouriercoefficients Ja

n are the same as those of the Jan ’s.

6.1 Wakimoto modules of critical level 163

Conversely, suppose that we are given vectors Ja−1|0〉V satisfying the con-

dition of the lemma. Define a linear map ρ : Vκ(g)→ V by the formula

Ja1n1. . . Jam

nm|0〉 7→ Ja1

n1. . . Jam

nm|0〉.

It is easy to check that this map is a homomorphism of Z-graded vertexalgebras.

The vertex algebra Mg is Z-graded (see Section 5.4.1). The following is acorollary of Theorem 5.6.8.

Corollary 6.1.2 There exists a homomorphism of Z-graded vertex algebrasVκc

(g)→Mg.

Proof. According to Theorem 5.6.8, there exists a homomorphism gκc→

U(Mg) such that 1 7→ 1 and

Ja(z) 7→ Y (Pa(a∗α,0, aβ,−1)|0〉, z) + Y (Ba, z),

where the vectors Pa and Ba have degree 1. Therefore by Lemma 6.1.1 weobtain a homomorphism of vertex algebras Vκc

(g)→Mg such that

Ja−1|0〉 7→ Pa(a∗α,0, aβ,−1)|0〉+Ba.

The complex C•loc(Lg,Ag0,loc) which we used to prove Theorem 5.6.8 carries

a gradation with respect to the root lattice of g defined by the formulas

wt a∗α,n = −α, wtψ∗a,n = −wt Ja, wt |0〉 = 0

so that wt∫Y (A, z)dz = wtA. The differential preserves this gradation, and

it is clear that the cocycles ω and i∗(σ) are homogeneous with respect to it.Therefore the element γ introduced in the proof of Theorem 5.6.8 may bechosen in such a way that it is also homogeneous with respect to the weightgradation. This means that Ba ∈ Mg,+ may be chosen in such a way thatwtBa = wtJa, in addition to the condition degBa = 1.

We then necessarily have Ba = 0 for all Ja ∈ h ⊕ n+, because there areno elements of such weights in Mg,+. Furthermore, we find that the term Ba

corresponding to Ja = fi must be proportional to a∗αi,−1|0〉 ∈ Mg,+. Usingthe formulas of Section 5.2.5 and the discussion of Section 5.3.2, we thereforeobtain a more explicit description of the homomorphism Vκc(g)→Mg:

Theorem 6.1.3 There exist constants ci ∈ C such that the Fourier coefficients


of the vertex operators

ei(z) = aαi(z) +∑

β∈∆+

:P iβ(a∗α(z))aβ(z):,

hi(z) = −∑

β∈∆+

β(hi):a∗β(z)aβ(z): ,

fi(z) =∑

β∈∆+

:Qiβ(a∗α(z))aβ(z): + ci∂za

∗αi

(z),

where the polynomials P iβ , Q

iβ are introduced in formulas (5.2.12)–(5.2.14),

generate an action of gκcon Mg.

In addition to the above homomorphism of Lie algebras wκc : gκc → Ag≤1,loc,

there is also a Lie algebra anti-homomorphism

wR : Ln+ → Ag≤1,loc

which is induced by the right action of n+ on N+ (see Section 5.2.5). Byconstruction, the images of Ln+ under wκc and wR commute. We have

wR(ei(z)) = aαi(z) +∑

β∈∆+

PR,iβ (a∗γ(z))aβ(z), (6.1.1)

where the polynomials PR,iβ were defined in Section 5.2.5. More generally, we

have

wκc(eα(z)) = aα(z) +∑

β∈∆+;β>α

Pαβ (a∗γ(z))aβ(z), (6.1.2)

wR(eα(z)) = aα(z) +∑

β∈∆+;β>α

PR,αβ (a∗γ(z))aβ(z), (6.1.3)

for some polynomials Pαβ and PR,α

β , where the condition β > α comes fromthe commutation relations

[h, eα] = α(h)eα, [h, eRα ] = α(h)eR

α , h ∈ h.

Note that there is no need to put normal ordering in the above three formulas,because Pα

β and PR,αβ cannot contain a∗β , for the same reason.

6.1.2 Other g-module structures on Mg

In Theorem 6.1.3 we constructed the structure of a gκc -module on Mg. Toobtain other gκc

-module structures of critical level on Mg, we need to considerother homomorphisms gκc

→ Ag≤1,loc lifting the homomorphism Lg → T g

loc.According to Lemma 5.5.3, the set of isomorphism classes of such liftingsis a torsor over H1(Lg,Ag

0,loc), which is the first cohomology of the “big”Chevalley complex


∧i(Lg),Ag

0,loc).

6.1 Wakimoto modules of critical level 165

Recall that our complex C•loc(Lg,Ag0,loc) has a weight gradation, and our

cocycle ω has weight 0. Therefore among all liftings we consider those whichhave weight 0. The set of such liftings is in bijection with the weight 0homogeneous component of H1(Lg,Ag

0,loc).

Lemma 6.1.4 The weight 0 component of H1(Lg,Ag0,loc) is isomorphic to the

(topological) dual space (Lh)∗ to Lh.

Proof. First we show that the weight 0 component of H1(Lg,Ag0,loc) is

isomorphic to the space of weight 0 cocycles in C1(Lg,Ag0,loc). Indeed, since

wt a∗α,n = −α, the weight 0 part of C0(Lg,Ag0,loc) = Ag

0,loc is one-dimensionaland consists of the constants. If we apply the differential to a constant, weobtain 0, and so there are no coboundaries of weight 0 in C1(Lg,Ag

0,loc).Next, we show that any weight 0 one-cocycle φ is uniquely determined by

its restriction to Lh ⊂ Lg, which may be an arbitrary (continuous) linearfunctional on Lh. Indeed, since the weights occurring in Ag

0,loc are less thanor equal to 0, the restriction of φ to L+n− is equal to 0, and the restriction toLh takes values in the constants C ⊂ Ag

0,loc. Now let us fix φ|Lh. We identify(Lh)∗ with h∗((t))dt using the residue pairing, and write φ|Lh as χ(t)dt usingthis identification. Here

χ(t) =∑n∈Z

χnt−n−1, χn ∈ h∗,

where χn(h) = φ(hn). We denote 〈χ(t), hi〉 by χi(t).We claim that for any χ(t)dt ∈ h∗((t))dt, there is a unique one-cocycle φ of

weight 0 in C1(Lg,Ag0,loc) such that

φ(ei(z)) = 0, (6.1.4)

φ(hi(z)) = χi(z), (6.1.5)

φ(fi(z)) = χi(z)a∗α,n(z). (6.1.6)

Indeed, having fixed φ(ei(z)) and φ(hi(z)) as in (6.1.4) and (6.1.5), we obtainusing the formula

φ ([ei,n, fj,m]) = ei,n · φ(fj,m)− fj,m · φ(ei,n)

that

δi,jχi,n+m = ei,n · φ(fj,m).

This equation on φ(fj,m) has a unique solution inAg0,loc of weight−αi, namely,

the one given by formula (6.1.6). The cocycle φ, if it exists, is uniquelydetermined once we fix its values on ei,n, hi,n, and fi,n. Let us show that it


exists. This is equivalent to showing that the Fourier coefficients of the fields

ei(z) = aαi(z) +∑

β∈∆+

:P iβ(a∗α(z))aβ(z):, (6.1.7)

hi(z) = −∑

β∈∆+

β(hi):a∗β(z)a∗β(z): + χi(z), (6.1.8)

fi(z) =∑

β∈∆+


∗αi

(z) + χi(z)a∗α,n(z), (6.1.9)

satisfy the relations of gκc with 1 = 1.Let us remove the normal ordering and set ci = 0, i = 1, . . . , `. Then the

corresponding Fourier coefficients are no longer well-defined as linear operatorson Mg. But they are well-defined linear operators on the space FunLU . Theresulting Lg-module structure is easy to describe. Indeed, the Lie algebra Lg

acts on FunLU by vector fields. More generally, for any Lb−-module R weobtain a natural action of Lg on the tensor product FunLU⊗R (this is justthe topological Lg-module induced from the Lb−-module R). If we choose asR the one-dimensional representation on which all fi,n act by 0 and hi,n actsby multiplication by χi,n for all i = 1, . . . , ` and n ∈ Z, then the correspondingLg-action on FunLU⊗R ' FunLU is given by the Fourier coefficients of theabove formulas, but with the normal ordering removed and ci = 0. Hence ifwe remove the normal ordering and set ci = 0, then these Fourier coefficientsdo satisfy the commutation relations of Lg.

When we restore the normal ordering, these commutation relations mayin general be distorted, due to the double contractions, as explained in Sec-tion 5.5.2. But we know from Theorem 6.1.3 that when we restore normalordering and set all χi,n = 0, then there exist the numbers ci such that theseFourier coefficients satisfy the commutation relations of gκc

with 1 = 1. Theterms (6.1.4)–(6.1.6) will not generate any new double contractions in thecommutators. Therefore the commutation relations of gκc are satisfied for allnon-zero values of χi,n. This completes the proof.

Corollary 6.1.5 For each χ(t) ∈ h∗((t)) there is a g-module structure ofcritical level on Mg, with the action given by formulas (6.1.7)–(6.1.9).

We call these modules the Wakimoto modules of critical level anddenote them by Wχ(t).

Let π0 be the commutative algebra C[bi,n]i=1,...,`;n<0 with the derivation Tgiven by the formula

T · bi1,n1 . . . bim,nm= −

m∑j=1

njbi1,n1 . . . bij ,nj−1 · · · bim,nm.

Then π0 is naturally a commutative vertex algebra (see Section 2.2.2). In

6.2 Deforming to other levels 167

particular, we have

Y (bi,−1, z) = bi(z) =∑n<0

bi,nz−n−1.

Using the same argument as in the proof of Lemma 6.1.4, we now obtain astronger version of Corollary 6.1.2.

Theorem 6.1.6 There exists a homomorphism of vertex algebras

wκc : Vκc(g)→Mg ⊗ π0

such that

ei(z) 7→ aαi(z) +∑

β∈∆+

:P iβ(a∗α(z))aβ(z):,

hi(z) 7→ −∑

β∈∆+

β(hi):a∗β(z)aβ(z): + bi(z),

fi(z) 7→∑

β∈∆+


∗αi

(z) + bi(z)a∗αi(z),


iβ are introduced in formulas (5.2.12)–(5.2.14).

Thus, any module over the vertex algebraMg⊗π0 becomes a Vκc(g)-module,and hence a g-module of critical level. We will not require that the moduleis necessarily Z-graded. In particular, for any χ(t) ∈ h∗((t)) we have a one-dimensional π0-module Cχ(t), on which bi,n acts by multiplication by χi,n. Thecorresponding gκc

-module is the Wakimoto module Wχ(t) introduced above.

6.2 Deforming to other levels

In this section we extend the construction of Wakimoto modules to non-criticallevels. We prove the existence of a homomorphism of vertex algebras

wκ : Vκ(g)→Mg ⊗ πκ−κc0 ,

where πκ−κc0 is the vertex algebra associated to the Heisenberg Lie algebra

hκ−κc, and analyze in detail the compatibility of this homomorphism with

the action of the group AutO of changes of coordinate on the disc. Thiswill enable us to give a coordinate-independent interpretation of the free fieldrealization.

6.2.1 Homomorphism of vertex algebras

As before, we denote by h the Cartan subalgebra of g. Let hκ be the one-dimensional central extension of the loop algebra Lh = h((t)) with the two-cocycle obtained by restriction of the two-cocycle on Lg corresponding to


the inner product κ. Then according to formula (1.3.4), hκ is a HeisenbergLie algebra. We will consider a copy of this Lie algebra with generatorsbi,n, i = 1, . . . , `, n ∈ Z, and 1 with the commutation relations

[bi,n, bj,m] = nκ(hi, hj)δn,−m1. (6.2.1)

Thus, the bi,n’s satisfy the same relations as the hi,n’s. Let πκ0 denote the

hκ-module induced from the one-dimensional representation of the abelianLie subalgebra of hκ spanned by bi,n, i = 1, . . . , `, n ≥ 0, and 1, on which 1acts as the identity and all other generators act by 0. We denote by |0〉 thegenerating vector of this module. It satisfies bi,n|0〉 = 0, n ≥ 0. Then πκ

0 hasthe following structure of a Z+-graded vertex algebra (see Theorem 2.3.7 of[FB]):

• Z+-grading: deg bi1,n1 . . . bim,nm |0〉 = −∑m

i=1 ni;• vacuum vector: |0〉;• translation operator: T |0〉 = 0, [T, bi,n] = −nbi,n−1;• vertex operators:

Y (bi,−1|0〉, z) = bi(z) =∑n∈Z

bi,nz−n−1,

Y (bi1,n1 . . . bim,nm |0〉, z) =n∏

j=1

1(−nj − 1)!

:∂−n1−1z bi1(z) . . . ∂

−nm−1z bim(z): .

The tensor product Mg ⊗ πκ−κc0 also acquires a vertex algebra structure.

The following theorem extends the result of Theorem 6.1.6 away from thecritical level.

Theorem 6.2.1 There exists a homomorphism of vertex algebras

wκ : Vκ(g)→Mg ⊗ πκ−κc0

such that

ei(z) 7→ aαi(z) +∑

β∈∆+

:P iβ(a∗α(z))aβ(z): ,

hi(z) 7→ −∑

β∈∆+

β(hi):a∗β(z)aβ(z): + bi(z), (6.2.2)

fi(z) 7→∑

β∈∆+

:Qiβ(a∗α(z))aβ(z): + (ci + (κ− κc)(ei, fi)) ∂za

∗αi



iβ were introduced in formulas (5.2.12)–(5.2.14).

Proof. Denote by Agloc the Lie algebra U(Mg ⊗ πκ−κc

0 ). By Lemma 6.1.1,in order to prove the theorem, we need to show that formulas (6.2.2) define


a homomorphism of Lie algebras gκ → Agloc sending the central element 1 to

the identity.Formulas (6.2.2) certainly define a linear map wκ : Lg → Ag

loc. Denote byωκ the linear map

∧2Lg→ Ag

loc defined by the formula

ωκ(f, g) = [wκ(f), wκ(g)]− wκ([f, g]).

Evaluating it explicitly in the same way as in the proof of Lemma 5.6.2, wefind that ωκ takes values in Ag

0,loc ⊂ Agloc. Furthermore, by construction of

wκ, for any X ∈ Ag0,loc and f ∈ Lg we have [wκ(f), X] = f ·X, where in the

right hand side we consider the action of f on the Lg-module Ag0,loc. This

immediately implies that ωκ is a two-cocycle of Lg with coefficients in Ag0,loc.

By construction, ωκ is local, i.e., belongs to C2loc(Lg,Ag

0,loc).Let us compute the restriction of ωκ to

∧2Lh. The calculations made in

the proof of Lemma 5.6.8 imply that

ωκ(hn, h′m) = n(κc(h, h′) + (κ− κc)(h, h′)) = nκ(h, h′).

Therefore this restriction is equal to the restriction of the Kac–Moody two-cocycle σκ on Lg corresponding to κ. Now Lemma 5.6.7 implies that thetwo-cocycle ωκ is cohomologically equivalent to i∗(σκ). We claim that it isactually equal to i∗(σκ).

Indeed, the difference between these cocycles is the coboundary of someelement γ ∈ C1

loc(Lg,Ag0,loc). The discussion before Theorem 6.1.3 implies

that γ(ei(z)) = γ(hi(z)) = 0 and γ(fi(z)) = c′i∂za∗αi

(z) for some constantsc′i ∈ C. In order to find the constants c′i we compute the value of the corre-sponding two-cocycle ωκ + dγ on ei,n and fi,−n. We find that it is equal tonσκ(ei,n, fi,−n)+c′in. On the other hand, the commutation relations in gκ re-quire that it be equal to nσκ(ei,n, fi,−n). Therefore c′i = 0 for all i = 1, . . . , `,and so γ = 0. Hence we have ωκ = i∗(σκ). This implies that formulas(6.2.2) indeed define a homomorphism of Lie algebras gκ → Ag

≤1,loc sendingthe central element 1 to the identity. This completes the proof.

Now any module over the vertex algebra Mg ⊗ πκ−κc0 becomes a Vκ(g)-

module and hence a gκ-module (with K acting as 1). For λ ∈ h∗, let πκ−κc

λ

be the Fock representation of hκ−κc generated by a vector |λ〉 satisfying

bi,n|λ〉 = 0, n > 0, bi,0|λ〉 = λ(hi)|λ〉, 1|λ〉 = |λ〉.

Then

Wλ,κdef= Mg ⊗ πκ−κc

λ

is an Mg⊗πκ−κc0 -module, and hence a gκ-module. We call it the Wakimoto

module of level κ and highest weight λ.


6.2.2 Wakimoto modules over sl2

In this section we describe explicitly the Wakimoto modules over sl2.Let e, h, f be the standard basis of the Lie algebra sl2. Let κ0 be the

invariant inner product on sl2 normalized in such a way that κ0(h, h) = 2. Wewill write an arbitrary invariant inner product κ on sl2 as kκ0, where k ∈ C,and will use k in place of κ in our notation. In particular, κc corresponds tok = −2. The set ∆+ consists of one element in the case of sl2, so we will dropthe index α in aα(z) and a∗α(z). Likewise, we will drop the index i in bi(z),etc., and will write M for Msl2 . We will also identify the dual space to theCartan subalgebra h∗ with C by sending χ ∈ h∗ to χ(h).

The Weyl algebra Asl2 has generators an, a∗n, n ∈ Z, with the commutation

relations

[an, a∗m] = δn,−m.

Its Fock representation is denoted by M . The Heisenberg Lie algebra hk hasgenerators bn, n ∈ Z, and 1, with the commutation relations

[bn, bm] = 2knδn,−m1,

and πkλ is its Fock representation generated by a vector |λ〉 such that

bn|λ〉 = 0, n > 0; b0|λ〉 = λ|λ〉; 1|λ〉 = |λ〉.

The module πk0 and the tensor product M ⊗ πk

0 are vertex algebras.The homomorphism wk : Vk(sl2) → M ⊗ πk+2

0 of vertex algebras is givenby the following formulas:

e(z) 7→ a(z)

h(z) 7→ −2:a∗(z)a(z): + b(z) (6.2.3)

f(z) 7→ −:a∗(z)2a(z): + k∂za∗(z) + a∗(z)b(z).

The Wakimoto module M ⊗ πk+2λ will be denoted by Wλ,k and its highest

weight vector will be denoted by |λ〉.We will use these explicit formulas in Chapter 7 in order to construct in-

tertwining operators between Wakimoto modules.

6.2.3 Conformal structures at non-critical levels

In this section we show that the homomorphism wκ of Theorem 6.2.1 is ahomomorphism of conformal vertex algebras when κ 6= κc (see Section 3.1.4for the definition of conformal vertex algebras). This will allow us to ob-tain a coordinate-independent version of this homomorphism. By taking thelimit κ → κc, we will also obtain a coordinate-independent version of thehomomorphism wκc

.


The vertex algebra Vκ(g), κ 6= κc, has the structure of a conformal vertexalgebra given by the Segal–Sugawara vector

Sκ =12

∑a

Ja−1Ja,−1|0〉, (6.2.4)

where Ja is the basis of g dual to the basis Ja with respect to the innerproduct κ − κc (see Section 3.1.4, where this vector was denoted by S). Weneed to calculate the image of Sκ under wκ.

Proposition 6.2.2 The image of Sκ under wκ is equal to ∑α∈∆+

aα,−1a∗α,−1 +

12

∑i=1

bi,−1bi−1 − ρ−2

|0〉, (6.2.5)

where bi is a dual basis to bi and ρ is the element of h corresponding toρ ∈ h∗ under the isomorphism induced by the inner product (κ− κc)|h.

The vector wκ(Sκ) defines the structure of a conformal algebra and hencean action of the Virasoro algebra on Mg ⊗ πκ−κ0

0 . The homomorphism wκ

intertwines the corresponding action of (DerO,AutO) on Mg ⊗ πκ−κ00 and

the natural action of (DerO,AutO) on Vκ(g).

Proof. The vertex algebras Vκ(g) and Mg ⊗ πκ−κc0 carry Z-gradings and

also weight gradings by the root lattice of g coming from the action of h (notethat wt aα,n = −wt a∗α,n = α,wt bi,n = 0). The homomorphism wκ preservesthese gradings. The vector Sκ ∈ Vκ(g) is of degree 2 with respect to theZ-grading and of weight 0 with respect to the root lattice grading. Hence thesame is true for wκ(Sκ).

The basis in the corresponding homogeneous subspace of Mg ⊗ πκ−κc0 is

formed by the monomials of the form

bi,−1bj,−1, bi,−2, aα,−1a∗α,−1, (6.2.6)

aα,−1aβ,−1a∗α,0a

∗β,0, aα,−1a

∗α,0bi,−1, (6.2.7)

aα,−2a∗α,0, aα+β,−2a

∗α,0a

∗β,0, aα+β,−1a

∗α,−1a

∗β,0, (6.2.8)

applied to the vacuum vector |0〉. We want to show that wκ(Sκ) is a linearcombination of the monomials (6.2.6).

By construction, the Fourier coefficients Ln, n ∈ Z, of the vertex operatorwκ(Sκ) preserve the weight grading on Mg⊗πκ−κc

0 , and we have degLn = −nwith respect to the vertex algebra grading on Mg⊗πκ−κc

0 . We claim that anyvector of the form P (a∗α,0)|0〉 is annihilated by Ln, n ≥ 0. This is clear forn > 0 for degree reasons.


To see that the same is true for L0, observe that

L0 · P (a∗α,0)|0〉 =12

∑a

wκ(Ja,0)wκ(Ja0 ) · P (a∗α,0)|0〉.

According to the formulas for the homomorphism wκ given in Theorem 6.2.1,the action of the constant subalgebra g ⊂ gκ on C[a∗α,0]α∈∆+ |0〉, obtained viawκ, coincides with the natural action of g on C[yα]α∈∆+ = FunN+, if we sub-stitute a∗α,0 7→ yα. Therefore the action of L0 on P (a∗α,0)|0〉 coincides with theaction of the Casimir operator 1

2

∑a JaJ

a on C[yα]α∈∆+ , under this substitu-tion. But as a g-module, the latter is the contragredient Verma module M∗0as we showed in Section 5.2.3. Therefore the action of the Casimir operatoron it is equal to 0.

The fact that all vectors of the form P (a∗α,0)|0〉 are annihilated by Ln, n ≥ 0,precludes the monomials (6.2.7) and the monomials (6.2.8), except for the lastone, from appearing in wκ(Sκ). In order to eliminate the last monomial in(6.2.8) we will prove that

Ln · aα,−1|0〉 = 0, n > 0 L0 · aα,−1|0〉 = aα,−1|0〉, α ∈ ∆+.

The first formula holds for degree reasons. To show the second formula, leteαα∈∆+ be a root basis of n+ ⊂ g such that eαi = ei. Then the vectorseα,−1|0〉 ∈ Vκ(g), and hence wκ(eα,−1|0〉) ∈ Mg ⊗ πκ−κc

0 are annihilated byLn, n > 0, and are eigenvectors of L0 with eigenvalue 1. We have

wκ(eα,−1|0〉) = aα,−1|0〉+∑

β∈∆+;β>α

Pαβ (a∗α,0)aβ,−1|0〉, (6.2.9)

where the polynomials Pαβ are found from the following formula for the action

of eα on U :

eα =∂

∂yα+

∑β∈∆+;β>α

Pαβ (yα)

∂

∂yβ

(see formula (6.1.2)).Starting from the maximal root αmax and proceeding by induction on de-

creasing heights of the roots, we derive from these formulas that L0 ·aα,−1|0〉 =aα,−1|0〉. This eliminates the last monomial in (6.2.8) and gives us the formula

wκ(Sκ) =∑

α∈∆+

aα,−1a∗α,−1|0〉 (6.2.10)

plus the sum of the first two types of monomials (6.2.6). It remains to deter-mine the coefficients with which they enter the formula.

In order to do that we use the following formula for wκ(hi,−1|0〉), whichfollows from Theorem 6.2.1:

wκ(hi,−1|0〉) =

− ∑β∈∆+

β(hi)a∗0,βaβ,−1 + bi,−1

|0〉. (6.2.11)


We find from it the action of L0 and L1 on the first summand of wκ(hi,−1|0〉):

L0 · −∑

β∈∆+

β(hi)a∗0,βaβ,−1|0〉 = −∑

β∈∆+

β(hi)a∗0,βaβ,−1|0〉,

L1 · −∑

β∈∆+

β(hi)a∗0,βaβ,−1|0〉 = −∑

β∈∆+

β(hi)|0〉 = −2ρ(hi)|0〉.

In addition, Ln, n > 1, act by 0 on it.On the other hand, we know that wκ(hi,−1|0〉) has to be annihilated by

Ln, n > 0, and is an eigenvector of L0 with eigenvalue 1. There is a uniquecombination of the first two types of monomials (6.2.6) which, when added to(6.2.10), satisfies this condition; namely,

12

∑i=1

bi,−1bi−1 − ρ−2. (6.2.12)

This proves formula (6.2.5). Now we verify directly, using Lemma 3.1.2, thatthe vector wκ(Sκ) given by this formula satisfies the axioms of a conformalvector.

This completes the proof.

6.2.4 Quasi-conformal structures at the critical level

Now we use this lemma to obtain additional information about the homomor-phism wκc

. Denote by O the complete topological ring C[[t]] and by DerOthe Lie algebra of its continuous derivations. Note that DerO ' C[[t]]∂t.

Recall that a vertex algebra V is called quasi-conformal if it carries anaction of the Lie algebra DerO satisfying the following conditions:

• the formula

[Lm, A(k)] =∑

n≥−1

(m+ 1n+ 1

)(Ln ·A)(m+k−n)

holds for all A ∈ V ;• the element L−1 = −∂t acts as the translation operator T ;• the element L0 = −t∂t acts semi-simply with integral eigenvalues;• the Lie subalgebra Der+O acts locally nilpotently

(see Definition 6.3.4 of [FB]). In particular, a conformal vertex algebra is au-tomatically quasi-conformal (with the DerO-action coming from the Virasoroaction).

The Lie algebra DerO acts naturally on gκc preserving g[[t]], and hence itacts on Vκc

(g). The DerO-action on Vκc(g) coincides with the limit κ → κc

of the DerO-action on Vκ(g), κ 6= κc, obtained from the Sugawara conformal


structure. Therefore this action defines the structure of a quasi-conformalvertex algebra on Vκc

(g).Next, we define the structure of a quasi-conformal algebra on Mg ⊗ π0

as follows. The vertex algebra Mg is conformal with the conformal vector(6.2.10), and hence it is also quasi-conformal. The commutative vertex algebraπ0 is the κ→ κc limit of the family of conformal vertex algebras πκ−κc

0 withthe conformal vector (6.2.12). The induced action of the Lie algebra DerOon πκ−κc

0 is well-defined in the limit κ→ κc and so it induces a DerO-actionon π0. Therefore it gives rise to the structure of a quasi-conformal vertexalgebra on π0. The DerO-action is in fact given by derivations of the algebrastructure on π0 ' C[bi,n], and hence by Lemma 6.3.5 of [FB] it defines thestructure of a quasi-conformal vertex algebra on π0. Explicitly, the action ofthe basis elements Ln = −tn+1∂t, n ≥ −1, of DerO on π0 is determined bythe following formulas:

Ln · bi,m = −mbi,n+m, −1 ≤ n < −m,Ln · bi,−n = n(n+ 1), n > 0, (6.2.13)

Ln · bi,m = 0, n > −m

(note that 〈ρ, hi〉 = 1 for all i). Now we obtain a quasi-conformal vertexalgebra structure on Mg ⊗ π0 by taking the sum of the above DerO-actions.

Since the quasi-conformal structures on Vκc(g) and Mg ⊗ π0 both arose asthe limits of conformal structures as κ→ κc, we obtain the following corollaryof Proposition 6.2.2:

Corollary 6.2.3 The homomorphism wκc: Vκc

→ Mg ⊗ π0 preserves quasi-conformal structures. In particular, it intertwines the actions of DerO andAutO on both sides.

6.2.5 Transformation formulas for the fields

We can now obtain the transformation formulas for the fields aα(z), a∗α(z) andbi(z) and the modules Mg and π0.

From the explicit formula (6.2.5) for the conformal vector wκ(Sκ) we findthe following commutation relations:

[Ln, aα,m] = −mam+n, [Ln, a∗α,m] = −(m− 1)a∗α,m−1.

In the same way as in Section 3.5.5 we derive from this that aα(z) transformsas a one-form on the punctured disc D× = Spec C((z)), while a∗α(z) transformsas a function on D×. In particular, we obtain the following description of themodule Mg.

Let us identify U with n+. Consider the Heisenberg Lie algebra Γ, which isa central extension of the commutative Lie algebra n+ ⊗ K ⊕ n∗+ ⊗ ΩK with


the cocycle given by the formula

f(t), g(t)dt 7→∫〈f(t), g(t)〉dt.

This cocycle is coordinate-independent, and therefore Γ carries natural actionsof DerO, which preserve the Lie subalgebra Γ+ = n+ ⊗ O ⊕ n∗+ ⊗ ΩO. Weidentify the completed Weyl algebra Ag with a completion of U(Γ)/(1 − 1),where 1 is the central element. The module Mg is then identified with theΓ-module induced from the one-dimensional representation of Γ+ ⊕ C1, onwhich Γ+ acts by 0, and 1 acts as the identity. The DerO-action on Mg

considered above is nothing but the natural action on the induced module.Now we consider the fields bi(z) =

∑n<0 bi,nz

−n−1 and the module π0.Formulas (6.2.13) describe the action of DerO on the bi,n’s and hence on theseries bi(z). In fact, these formulas imply that ∂z + bi(z) transforms as aconnection on the line bundle Ω−〈ρ,hi〉.

More precisely, let LH be the dual group to H, i.e., it is the complex torusthat is determined by the property that its lattice of characters LH → C× isthe lattice of cocharacters C× → H, and the lattice of cocharacters of LH isthe lattice of characters of H.† The Lie algebra Lh of LH is then canonicallyidentified with h∗. Denote by Ω−ρ the unique principal LH-bundle on thedisc D such that the line bundle associated to any character λ : LH → C×(equivalently, a cocharacter of H) is Ω−〈ρ,λ〉 (see Section 4.2.3). Denote byConn(Ω−ρ)D the space of all connections on this LH-bundle. This is a torsorover h∗ ⊗ ΩO.

The above statement about bi(z) may be reformulated as follows: considerthe h∗-valued field b(z) =

∑ì=1 bi(z)ωi such that 〈b(z), hi〉 = bi(z). Then the

operator ∂z +b(z) transforms as a connection on the LH-bundle Ω−ρ over D.Equivalently, ∂z +bi(z) transforms as a connection on the line bundle Ω−〈ρ,hi〉

over D.To see that, let w be a new coordinate such that z = ϕ(w); then the same

connection will appear as ∂w + b(w), where

b(w) = ϕ′ · b(ϕ(w)) + ρϕ′′

ϕ′. (6.2.14)

It is straightforward to check that formula (6.2.14) is equivalent to formula(6.2.13).

This implies that π0 is isomorphic to the algebra Fun(Conn(Ω−ρ)D) of func-tions on the space Conn(Ω−ρ)D. If we choose a coordinate z on D, then weidentify Conn(Ω−ρ)D with h∗⊗ΩO this algebra with the free polynomial alge-bra C[bi,n]i=1,...,`;n<0, whose generators bi,n are the following linear functionals

† Thus, LH is the Cartan subgroup of the Langlands dual group LG of G introduced inSection 1.1.5.


on h∗ ⊗ ΩO ' h∗[[z]]dz:

bi,n(χ(z)dz) = Resz=0〈χ(z), hi〉zndz.

6.2.6 Coordinate-independent version

Up to now, we have considered Wakimoto modules as representations of theLie algebra gκc

, which is the central extension of Lg = g((t)). As explained inSection 3.5.2, it is important to develop a theory which applies to the centralextension gκ,x of the Lie algebra g(Kx) = g⊗Kx, where Kx is the algebra offunctions on the punctured disc around a point x of a smooth curve. In otherwords, Kx is the completion of the field of functions on X corresponding to x.It is a topological algebra which is isomorphic to C((t)), but non-canonically.If we choose a formal coordinate t at x, we may identify Kx with C((t)), butthis identification is non-canonical as there is usually no preferred choice ofcoordinate t at x. However, as we saw in Section 3.5.2, we have a canonicalcentral extension gκc,x of g(Kx) and the vacuum gκc,x-module

Vκ(g)x = Indbgκ,x

g⊗Ox⊕CK C.

where, as before, Ox denotes the ring of integers in Kx which is isomorphicto C[[t]].

As explained in Section 3.5.4, Vκ(g)x may also be obtained as the twist ofVκ(g) by the AutO-torsor Autx of formal coordinates at x:

Vκ(g)x = Autx ×AutO

Vκ(g).

Now we wish to recast the free field realization homomorphism

wκ : Vκ(g)→Mg ⊗ πκ−κc0

of Theorem 6.2.1 in a coordinate-independent way.We have an analogue of the Heisenberg Lie algebra Γ introduced in the

previous section, attached to a point x. By definition, this Lie algebra, denotedby Γx, is the central extension of (n+ ⊗ Kx) ⊕ (n∗+ ⊗ ΩKx

). Let Γ+,x be itscommutative Lie subalgebra (n+ ⊗ Ox) ⊕ (n∗+ ⊗ ΩOx

). Let Mg,x be the Γx-module

Mg,x = IndΓx

Γ+,x⊕C1 C.

As in the case of Vκ(g)x, we have

Mg,x = Autx ×AutO

Mg,

where the action of AutO on Mg comes from the action of coordinate changeson Mg induced by its action on the Lie algebra Γ, as described in the previoussection.


Next, we need a version of the Heisenberg Lie algebra hν and its moduleπν

0 attached to the point x.Consider the vector space Connλ(Ω−ρ)D×

xof λ-connections on the LH-

bundle Ω−ρ on D×x = SpecKx, for all possible complex values of λ. If wechoose an isomorphism Kx ' C((t)), then a λ-connection is an operator ∇ =λ∂t + χ(t), where χ(t) ∈ h∗((t)). We have an exact sequence

0→ h∗ ⊗ ΩKx → Connλ(Ω−ρ)D×x→ C∂t → 0,

where the penultimate map sends ∇ as above to λ∂t.Let hν,x be the topological dual vector space to Connλ(Ω−ρ)D×

x. It fits

into an exact sequence (here we use the residue pairing between ΩKx and Kx)

0→ C1→ hν,x → h⊗Kx → 0,

where 1 is the element dual to ∂x. The Lie bracket is given by the old formula(see Section 6.2.1); it is easy to see that this formula (which depends on ν) iscoordinate-independent.

Note that this sequence does not have a natural coordinate-independentsplitting. However, there is a natural splitting h ⊗ Ox → hν,x (the image isthe orthogonal complement to the space of λ-connections on the disc Dx).Therefore we define an hν,x-module

πνλ,x = Ind

bhν,x

h⊗Ox⊕C1 λ.

From the description of the action of AutO on the Lie algebra hν obtainedin the previous section it follows immediately that

πνλ,x = Autx ×

AutOπν

λ.

Now we can state a coordinate-independent version of the free field homo-morphism wκ.

Proposition 6.2.4 For any level κ and any point x there is a natural ho-momorphism of gκ,x-modules Vκ(g)x → Mg,x ⊗ πκ−κc

0,x . Furthermore, for anyhighest weight λ ∈ h∗,

Wλ,κ,x = Mg,x ⊗ πκ−κc0,x

carries a natural structure of gκ,x-module.

Proof. According to Proposition 6.2.2 and Corollary 6.2.3, the map wκ,considered as a homomorphism of gκ,x-modules, is compatible with the ac-tion of AutO on both sides. Therefore we may twist it by the torsor Autx.According to the above discussion, we obtain the first assertion of the propo-sition. Likewise, twisting the action of g on Wλ,κ by Autx, we obtain thesecond assertion.


In particular, for ν = 0 the Lie algebra h0,x is commutative. Any con-nection ∇ on Ω−ρ over the punctured disc D×x defines a linear functional onh0,x, and hence a one-dimensional representation C∇ of h0,x, considered as acommutative Lie algebra. If we choose an isomorphism Kx ' C((t)), then theconnection is given by the formula ∇ = ∂t + χ(t), where χ(t) ∈ h∗((t)). Theaction of the generators bi,n on C∇ is then given by the formula

bi,n 7→∫〈χ(z), hi〉zndz.

By Theorem 6.1.6, there exists a homomorphism of vertex algebras wκc:

Vκc(g)→Mg⊗π0. According to Corollary 6.2.3, it commutes with the action

of AutO on both sides. Therefore the corresponding homomorphism of Liealgebras g→ U(Mg⊗ π0) also commutes with the action of AutO. Hence wemay twist this homomorphism with the AutO-torsor Autx. Then we obtaina homomorphism of Lie algebras

gκc,x → U(Mg ⊗ π0)xdef= Autx ×

AutOU(Mg ⊗ π0).

Let us call a Γx ⊕ h0,x-module smooth if any vector in this module isannihilated by the Lie subalgebra

(n+ ⊗mNx )⊕ (n∗+ ⊗mN

x ΩO)⊕ (h⊗mNx ),

where m is the maximal ideal of Ox, for sufficiently large N ∈ Z+. Clearly,any smooth Γ⊕ h0-module is automatically a U(Mg⊗π0)-module. Hence anysmooth Γx ⊕ h0,x-module is automatically a U(Mg ⊗ π0)x-module and hencea gκc,x-module.

Proposition 6.2.5 For any connection on the LH-bundle Ω−ρ over the punc-tured disc D×x = SpecKx there is a canonical gκc,x-module structure on Mg,x.

Proof. Note that Mg,x is a smooth Γx-module, and C∇ is a smooth h0,x-module for any connection ∇ on the LH-bundle Ω−ρ over the punctured discD×x . Taking the tensor product of these two modules we obtain a gκc,x-module, which is isomorphic to Mg,x as a vector space.

Thus, we obtain a family of gκc,x-modules parameterized by the connectionson the LH-bundle Ω−ρ over the punctured disc D×x .

If we choose an isomorphism Kx ' C((t)) and write a connection ∇ as ∇ =∂t + χ(t), then the module corresponding to ∇ is nothing but the Wakimotomodule Wχ(z) introduced in Corollary 6.1.5.

6.3 Semi-infinite parabolic induction

In this section we generalize the construction of Wakimoto modules by con-sidering an arbitrary parabolic subalgebra of g instead of a Borel subalgebra.

6.3 Semi-infinite parabolic induction 179

6.3.1 Wakimoto modules as induced representations

The construction of the Wakimoto modules presented above may be summa-rized as follows: for each representation N of the Heisenberg Lie algebra hκ,we have constructed a gκ+κc

-module structure on Mg ⊗ N . The procedureconsists of extending the hκ-module by 0 to b−,κ, followed by what may beviewed as a semi-infinite analogue of induction functor from b−,κ-modules togκ+κc

-modules. An important feature of this construction, as opposed to theordinary induction, is that the level gets shifted by κc. In particular, if westart with an h0-module, or, equivalently, a representation of the commutativeLie algebra Lh, then we obtain a gκc

-module of critical level, rather than oflevel 0. For instance, we can apply this construction to irreducible smooth rep-resentations of the commutative Lie algebra Lh. These are one-dimensionaland are in one-to-one correspondence with the elements χ(t) of the (topo-logical) dual space (Lh)∗ ' h∗((t))dt. As the result we obtain the Wakimotomodules Wχ(t) of critical level introduced above. Looking at the transforma-tion properties of χ(t) under the action of the group AutO of changes of thecoordinate t, we find that χ(t) actually transforms not as a one-form, but asa connection on a specific LH-bundle. This “anomaly” is a typical feature of“semi-infinite” constructions.

In contrast, if κ 6= 0, the irreducible smooth hκ-modules are just the Fockrepresentations πκ

λ. To each of them we attach a gκ+κc -module Wλ,κ+κc .Now we want to generalize this construction by replacing the Borel sub-

algebra b− and its Levi quotient h by an arbitrary parabolic subalgebra p

and its Levi quotient m. Then we wish to attach to a module over a centralextension of the loop algebra Lm a g-module. It turns out that this is indeedpossible provided that we pick a suitable central extension of Lm. We callthe resulting g-modules the generalized Wakimoto modules corresponding top. Thus, we obtain a functor from the category of smooth m-modules to thecategory of smooth g-modules. It is natural to call it the functor of semi-infinite parabolic induction (by analogy with a similar construction forrepresentations of reductive groups).

6.3.2 The main result

Let p be a parabolic Lie subalgebra of g. We will assume that p contains thelower Borel subalgebra b− (and so, in particular, p contains h ⊂ b−). Let

p = m⊕ r (6.3.1)

be a Levi decomposition of p, where m is a Levi subgroup containing h and r

is the nilpotent radical of p. Further, let

m =s⊕

i=1

mi ⊕m0


be the direct sum decomposition of m into the direct sum of simple Lie sub-algebras mi, i = 1, . . . , s, and an abelian subalgebra m0 such that these directsummands are mutually orthogonal with respect to the inner product on g.We denote by κi,c the critical inner product on mi, i = 1, . . . , s, defined as inSection 5.3.3. We also set κ0,c = 0.

Given a set of invariant inner products κi on mi, 0 = 1, . . . , s, we obtainan invariant inner product on m. Let m(κi) be the corresponding affine Kac–Moody algebra, i.e., the one-dimensional central extension of Lm with thecommutation relations given by formula (1.3.4). We denote by Vκi(mi), i =1, . . . , s, the vacuum module over mi,κi

with the vertex algebra structure de-fined as in Section 6.1.1. We also denote by Vκ0(m0) the Fock representationπκ0

0 of the Heisenberg Lie algebra mκ0 with its vertex algebra structure definedas in Section 6.2.1. Let

V(κi)(m) def=s⊗

i=0

Vκi(mi)

be the vacuum module over m(κi) with the tensor product vertex algebrastructure.

Denote by ∆′+ the set of positive roots of g which do not occur in theroot space decomposition of p (note that by our assumption that b− ⊂ p,all negative roots do occur). Let Ag,p be the Weyl algebra with generatorsaα,n, a

∗α,n, α ∈ ∆′+, n ∈ Z, and relations (5.3.3). Let Mg,p be the Fock repre-

sentation of Ag,p generated by a vector |0〉 such that

aα,n|0〉 = 0, n ≥ 0; a∗α,n|0〉 = 0, n > 0.

Then Mg,p carries a vertex algebra structure defined as in Section 5.4.1.We have the following analogue of Theorem 6.2.1.

Theorem 6.3.1 Suppose that κi, i = 0, . . . , s, is a set of inner products suchthat there exists an inner product κ on g whose restriction to mi equals κi−κi,c

for all i = 0, . . . , s. Then there exists a homomorphism of vertex algebras

wpκ : Vκ+κc(g)→Mg,p ⊗ V(κi)(m).

Proof. The proof is a generalization of the proof of Theorem 6.2.1 (in fact,Theorem 6.2.1 is a special case of Theorem 6.3.1 when p = b−). Let P be theLie subgroup of G corresponding to p, and consider the homogeneous spaceG/P . It has an open dense subset Up = Np · [1], where Np is the subgroup ofN+ corresponding to the subset ∆′+ ⊂ ∆+. We identify Up with Np and withits Lie algebra np using the exponential map.

Set LUp = Up((t)). We define functions and vector fields on LUp, denotedby FunLUp and VectLUp, respectively, in the same way as in Section 5.3.1.The action of Lg on Up((t)) gives rise to a Lie algebra homomorphism Lg →


VectLUp, in the same way as before. We generalize this homomorphism asfollows.

Consider the quotient G/R, where R is the Lie subgroup of G correspondingto the nilpotent Lie algebra r appearing in the Levi decomposition (6.3.1).We have a natural projection G/R → G/P , which is an M -bundle, whereM = P/R is the Levi subgroup of G corresponding to m. Over Up ⊂ G/P

this bundle may be trivialized, and so it is isomorphic to Up ×M . The Liealgebra g acts on this bundle, and hence on Up×M . As the result, we obtaina Lie algebra homomorphism g → Vect(Up ×M). It is easy to see that itfactors through homomorphisms

g→ (VectUp ⊗ 1)⊕ (FunUp ⊗m)

and m→ VectM .The loop version of this construction gives rise to a (continuous) homomor-

phism of (topological) Lie algebras

Lg→ VectLUp ⊕ FunUp⊗Lm.

Moreover, the image of this homomorphism is contained in the “local part,”i.e., the direct sum of the local part T g,p

loc of VectLUp defined as in Section 5.4.3and the local part Ig,p

loc of FunUp⊗Lm. By definition, Ig,ploc is the span of the

Fourier coefficients of the formal power series P (∂nz a∗α(z))Ja(z), where P is a

differential polynomial in a∗α(z), α ∈ ∆′+, and Ja ∈ m.Let Ag,p

0,loc and Ag,p≤1,loc be the zeroth and the first terms of the natural

filtration on the local completion of the Weyl algebra Ag,p, defined as inSection 5.3.3. We have a non-split exact sequence of Lie algebras

0→ Ag,p0,loc → A

g,p≤1,loc → T

g,ploc → 0. (6.3.2)

Set

J g,ploc

def= Ag,p≤1,loc ⊕ I

g,ploc , (6.3.3)

and note that J g,ploc is naturally a Lie subalgebra of the local Lie algebra

U(Mg,p ⊗ V(κi)(m)). Using the splitting of the sequence (6.3.2) as a vectorspace via the normal ordering, we obtain a linear map w(κi) : Lg→ J g,p

loc .We need to compute the failure of w(κi) to be a Lie algebra homomorphism.

Thus, we consider the corresponding linear map ω(κi) :∧2

Lg→ J g,ploc defined

by the formula

ω(κi)(f, g) = [wκ(f), wκ(g)]− wκ([f, g]).

Evaluating it explicitly in the same way as in the proof of Lemma 5.6.2, wefind that ω(κi) takes values in Ag,p

0,loc ⊂ Jg,ploc . Furthermore, Ag,p

0,loc is naturallyan Lg-module, and by construction of w(κi), for any X ∈ Ag,p

0,loc and f ∈ Lg

we have [w(κi)(f), X] = f · X. This implies that ω(κi) is a two-cocycle of


Lg with coefficients in Ag,p0,loc. By construction, it is local, i.e., belongs to

C2loc(Lg,Ag,p

0,loc) (defined as in Section 5.6.2).Following the argument used in the proof of Lemma 5.6.7, we show that

any two cocycles in C2loc(Lg,Ag,p

0,loc), whose restrictions to∧

2(Lm) coincide,represent the same cohomology class.

Let us compute the restriction of ω(κi) to∧2

Lm. For this we evaluate w(κi)

on elements of Lm. Let Jαα∈∆′+

be a basis of np. The adjoint action of theLie algebra m on g preserves np, and so we obtain a representation ρnp of m

on np. For any element A ∈ m we have

ρnp(A) · Jα =∑

β∈∆′+

cαβ(A)Jβ

for some cαβ(A) ∈ C. Therefore we obtain the following formula:

w(κi)(A(z)) = −∑

β∈∆′+

cαβ(A):a∗β(z)aβ(z): + A(z), A ∈ m, (6.3.4)

where A(z) =∑

n∈Z(A ⊗ tn)z−n−1, considered as a generating series of ele-ments of m ⊂ Ig,p

loc .Let κnp be the inner product on m defined by the formula

κnp(A,B) = Trnp ρnp(A)ρnp(B).

Computing directly the commutation relations between the coefficients of theseries (6.3.4), we find that for A ∈ mi, B ∈ mj we have the following formulas:

ω(κi)(An, Bm) = −nκnp(A,B)δn,−m, (6.3.5)

if i 6= j, and

ω(κi)(An, Bm) = n(−κnp(A,B) + κi(A,B))δn,−m, (6.3.6)

if i = j. Thus, the restriction of ω(κi) to∧2

Lm takes values in the subspaceof constants C ⊂ Ag,p

0,loc.Let κg be the Killing form on g and κmi the Killing form on mi (in particular,

κm0 = 0). Then we have

κg(A,B) = κmi(A,B) + 2κnp(A,B), if i = j, (6.3.7)

κg(A,B) = 2κnp(A,B), if i 6= j.

The factor of 2 is due to the fact that we have to include both positive andnegative roots. Recall our assumption that mi and mj are orthogonal withrespect to κg for all i 6= j. This implies that κnp(A,B) = 0, if i 6= j.

Recall that by definition κc = − 12κg, and κi,c = − 1

2κmi. Hence formula

(6.3.7) implies that

κnp |mi = −κc + κi,c.


Therefore we find that if κi = κ|mi + κi,c for some invariant inner product κon g, then

−κnp |mi+ κi = (κ+ κc)|mi

.

By inspection of formulas (6.3.5) and (6.3.6), we now find that if κ is aninvariant inner product on g whose restriction to mi equals κi − κi,c for alli = 0, . . . , s, then the restriction of the two-cocycle ω(κi) to

∧2(Lm) is equal to

the restriction to∧

2(Lm) of the two-cocycle σκ+κc on Lg (this is the cocyclerepresenting the one-dimensional central extension of g corresponding to theinner product κ+ κc on g).

Applying the argument used in the proof of Lemma 5.6.7, we find thatunder the above conditions, which are precisely the conditions stated in thetheorem, the two-cocycle ω(κi) on Lg is equivalent to the two-cocycle σκ+κc

.Therefore we obtain, in the same way as in the proof of Theorem 5.6.8, that

under these conditions the linear map w(κi) : Lg → J g,ploc may be modified

by the addition of an element of C1loc(Lg,Ag,p

0,loc) to give us a Lie algebrahomomorphism

gκ+κc→ J g,p

loc ⊂ U(Mg,p ⊗ V(κi)(m)).

Now Lemma 6.1.1 implies that there exists a homomorphism of vertex algebras

wpκ : Vκ+κc(g)→Mg,p ⊗ V(κi)(m).

This completes the proof.

Let us call an m(κi)-module smooth if any vector in it is annihilated bythe Lie subalgebra m⊗ tNC[[t]] for sufficiently large N .

Corollary 6.3.2 For any smooth m(κi)-module R with the κi’s satisfying theconditions of Theorem 6.3.1, the tensor product Mg,p⊗R is naturally a smoothgκ+κc

-module. There is a functor from the category of smooth m(κi)-modulesto the category of smooth gκ+κc

-modules sending a module R to Mg,p⊗R andm(κi)-homomorphism R1 → R2 to the gκ+κc-homomorphism Mg,p ⊗ R1 →Mg,p ⊗R2.

We call the gκ+κc-module Mg,p⊗R the generalized Wakimoto module

corresponding to R.Consider the special case when R is the tensor product of the Wakimoto

modules Wλi,κi over mi, i = 1, . . . , s, and the Fock representation πκ0λ0

overthe Heisenberg Lie algebra m0. In this case it follows from the constructionthat the corresponding gκ+κc

-module Mg,p⊗R is isomorphic to the Wakimotomodule Wλ,κ+κc

over gκ+κc, where λ = (λi).

Finally, let us suppose that the κi’s are chosen in such a way that theconditions of Theorem 6.3.1 are not satisfied. Then the two-cocycle ω(κi) onLg with coefficients in Ag,p

0,loc defined in the proof of Theorem 6.3.1, restricted


to Lm, still gives rise to a two-cocycle of Lm with coefficients in C. However,this two-cocycle is no longer equivalent to the restriction to Lm of any two-cocycle σ on Lg (those can be represented by the cocycles σν correspondingto invariant inner products ν on g). Using the same argument as in the proofof Lemma 5.6.7, we obtain that the two-cocycle ω(κi) on Lg with coefficientsin Ag,p

0,loc cannot be equivalent to a two-cocycle on Lg with coefficients inC ⊂ Ag,p

0,loc. Therefore the map w(κi) : Lg → J g,ploc cannot be lifted to a Lie

algebra homomorphism gν → J g,ploc (for any ν) in this case. In other words,

the conditions of Theorem 6.3.1 are the necessary and sufficient conditions forthe existence of such a homomorphism.

6.3.3 General parabolic subalgebras

So far we have worked under the assumption that the parabolic subalgebra p

contains b−. It is also possible to construct Wakimoto modules associated toother parabolic subalgebras. Let us explain how to do this in the case whenp = b+.

Let N be any module over the vertex algebra Mg ⊗ πκ−κc0 . There is an

involution of g sending ei to fi and hi to −hi. Under this involution b− goesto b+. This involution induces an involution on gκ. Then Theorem 6.2.1implies that the following formulas define a gκ-structure on N (with 1 actingas the identity):

fi(z) 7→ aαi(z) +∑

β∈∆+

:P iβ(a∗α(z))aβ(z): ,

hi(z) 7→∑

β∈∆+

β(hi):a∗β(z)aβ(z):− bi(z),

ei(z) 7→∑

β∈∆+

:Qiβ(a∗α(z))aβ(z): + (ci + (κ− κc)(ei, fi)) ∂za

∗αi



iβ were introduced in formulas (5.2.12)–(5.2.14).

For the resulting gκ-module to be a module with highest weight, we chooseN as follows. Let M ′g be the Fock representation of the Weyl algebra Ag

generated by a vector |0〉′ such that

aα,n|0〉′ = 0, n > 0; a∗α,n|0〉′ = 0, n ≥ 0.

We take as N the module M ′g ⊗ πκ−κc

−2ρ−λ, where πκ−κc

−2ρ−λ is the πκ−κc0 -module

defined in Section 6.2.1. We denote this module by W+λ,κ. This is the gener-

alized Wakimoto module corresponding to the parabolic subalgebra b+. Wewill use the same notation |0〉′ for the vector

|0〉′ ⊗ | − 2ρ− λ〉 ∈M ′g ⊗ πκ−κc

−2ρ−λ = W+λ,κ. (6.3.8)

The following result, which identifies a particular Wakimoto module witha Verma module, will be used in Section 8.1.1.


Consider the Lie algebra n+ = (g ⊗ tC[[t]]) ⊕ (n+ ⊗ 1). For λ ∈ h∗, letCλ be the one-dimensional representation of n+ ⊕ (h ⊗ 1) ⊕ C1, on whichn+ = (g⊗ tC[[t]])⊕ (n+⊗1) acts by 0, h⊗1 acts according to λ, and 1 acts asthe identity. Define the Verma module Mλ,κ of level κ and highest weightλ as the corresponding induced gκ module:

Mλ,κ = Indbgκbn+⊕(h⊗1)⊕1Cλ. (6.3.9)

The image in Mλ,κ of the vector 1 ⊗ 1 of this tensor product is the highestweight vector of Mλ,κ. We denote it by vλ,κ.

Proposition 6.3.3 The Wakimoto module W+0,κc

is isomorphic to the Vermamodule M0,κc

.

Proof. The vector |0〉′ ∈W+0,κc

given by formula (6.3.8) satisfies the sameproperties as the highest weight vector vλ,κ ∈ M0,κc

: it is annihilated by n+,the Lie algebra h ⊗ 1 acts via the functional λ, and 1 acts as the identity.Therefore there is a non-zero homomorphism M0,κc → W+

0,κcsending the

highest weight vector of M0,κcto |0〉′ ∈W+

0,κc.

We start by showing that the characters of the modules M0,κc and W+0,κc

areequal. This will reduce the problem to showing that the above homomorphismis surjective.

Let us recall the notion of character of a gκ-module. Suppose that we havea gκ-module M equipped with an action of the grading operator L0 = −t∂t,compatible with its action on gκ.†

Suppose in addition that L0 and h ⊗ 1 ⊂ gκ act diagonally on M withfinite-dimensional common eigenspaces. Then we define the character of Mas the formal series

chM =∑

bλ∈(h⊕CL0)∗

dimM(λ) ebλ, (6.3.10)

where M(λ) is the generalized eigenspace of L0 and h ⊗ 1 corresponding toλ : (h⊕ CL0)∗ → C.

The direct sum (h ⊗ 1) ⊕ CL0 ⊕ C1 is in fact the Cartan subalgebra ofthe extended Kac–Moody algebra g′κ = CL0 n gκ (see [K2]). Elements ofthe dual space to this Cartan subalgebra are called weights. We will considerthe weights occurring in modules on which the central element 1 acts as theidentity. Therefore without loss of generality we may view these weights aselements of the dual space to h = (h ⊗ 1) ⊕ CL0, and hence as pairs (λ, φ),where λ ∈ h∗ and φ is the value of −L0 = t∂t. We will use the standardnotation δ = (0, 1).

† Note that if κ 6= κc, then any smooth bgκ-module carries an action of the Virasoroalgebra obtained via the Segal–Sugawara construction, and so in particular an L0 action.However, general bgκc -modules do not necessarily carry an L0 action.


The set of positive roots of g is naturally a subset of h∗:

∆+ = α+ nδ |α ∈ ∆+, n ≥ 0 t −α+ nδ |α ∈ ∆+, n > 0 t nδ |n > 0.

The roots of the first two types are real roots; they have multiplicity 1. Theroots of the last type are imaginary; they have multiplicity `.

We have a natural partial order on the set h∗ of weights: λ > µ if λ− µ =∑i βi, where the βi’s are positive roots.Let Mλ,κ be the Verma module over gκ defined above. There is a unique

way to extend the action of gκ to g′κ by setting L0 · vλ,κc = 0 and using thecommutation relations [L0, An] = −nAn between L0 and gκ to define theaction of L0 on the rest of Mλ,κ. The resulting module is the Verma moduleover g′κc

with highest weight λ = (λ, 0), which we will denote by Mbλ,κ.By the Poincare–Birkhoff–Witt theorem, as a vector space Mbλ,κ is isomor-

phic to U(n−), where n− = (g ⊗ t−1C[t−1]) ⊕ (n− ⊗ 1). Therefore we obtainthe following formula for the character of Mbλ,κ:

ch Mbλ,κ = ebλ ∏

bα∈b∆+

(1− e−bα)−mult bα, (6.3.11)

where ∆+ is the set of positive roots of gκ.On the other hand, we have an action of DerO, and in particular of L0, on

W+0,κc

coming from the quasi-conformal vertex algebra structure onMg⊗π0 de-scribed in Section 6.2.4. It follows from the formulas obtained in Section 6.2.5that we have the following commutation relations:

[L0, aα,n] = −naα,n, [L0, a∗α,n] = −na∗α,n, [L0, bi,n] = −nbi,n.

These formulas, together with the requirement that L0|0〉′ = 0, uniquelydetermine the action of L0 on W+

0,κc. Next, we have an action of the Cartan

subalgebra h⊗ 1 ⊂ gκcon W+

0,κcsuch that

[h, aα,n] = α(h)aα,n, [h, a∗α,n] = α(h)a∗α,n, [h, bi,n] = 0

for h ∈ h. Both L0 and h ⊗ 1 act by 0 on |0〉 ∈ W+0,κc

. Since W+0,κc

hasa basis of monomials in aαn

, α ∈ ∆+, n < 0; a∗α,n, α ∈ ∆+, n ≤ 0; andbi,n, i = 1, . . . , `, n < 0, we find that the character of W+

0,κcis equal to the

character of M0,κcgiven by formula (6.3.11).

The homomorphism M0,κc → W+0,κc

intertwines the action of g′κc= CL0 n

gκc on both modules. Therefore our proposition will follow if we show thatthe homomorphism M0,κc

→ W+0,κc

is surjective, or, equivalently, that W+0,κc

is generated by the vector |0〉′ given by formula (6.3.8).Suppose that W+

0,κcis not generated by |0〉′. Then the space of coinvariants

of W+0,κc

with respect to the Lie algebra

n− = (n− ⊗ 1)⊕ (g⊗ t−1C[t−1])


has dimension greater than 1, because it must include some vectors in additionto the one-dimensional subspace spanned by the image of the highest weightvector. This means that there exists a homogeneous linear functional onW+

0,κc, whose weight is less than the highest weight (0, 0) and which is n−-

invariant.Then it is in particular invariant under the Lie subalgebra

L−b− = n− ⊗ C[t−1]⊕ h⊗ t−1C[t−1].

Therefore this functional necessarily factors through the space of coinvariantsof W+

0,κcby L−b−.

However, it follows from the construction of W+0,κc

that L−b− acts freely onW+

0,κc, and the space of coinvariants with respect to this action is isomorphic

to the subspace

C[a∗α,n]α∈∆+,n<0 ⊂W+0,κc

.

Indeed, it follows from the explicit formulas (6.1.2) and (6.2.2) for the ac-tion of eα(z) and hi(z) (which become fα(z) and −hi(z) after we apply ourinvolution) that the lexicographically ordered monomials∏

la<0

hia,la

∏mb≤0

fαb,mb

∏nc<0

a∗βc,nb|0〉′

form a basis of W+0,κc

.Hence we obtain that any L−b−-invariant functional on W+

0,κcis completely

determined by its restriction to the subspace C[a∗α,n]α∈∆+,n<0. Thus, a non-zero L−b−-invariant functional on W+

0,κcof weight strictly less than the high-

est weight (0, 0) necessarily takes a non-zero value on a homogeneous subspaceof C[a∗α,n]α∈∆+,n<0 of non-zero weight. But the weights of these subspacesare of the form

−∑

j

(njδ − βj), nj > 0, βj ∈ ∆+. (6.3.12)

Since the weight of our subspace is supposed to be less than the highest weightby our assumption, the number of summands in this formula has to be non-zero.

This implies that W+0,κc

must have an irreducible subquotient of highestweight of this form. Since the characters of W+

0,κcand M0,κc

coincide, andthe characters of irreducible highest weight representations are linearly inde-pendent (see [KK]), we find that M0,κc also has an irreducible subquotient ofhighest weight of the form (6.3.12).

Now recall the Kac–Kazhdan theorem [KK] describing the set of highestweights of irreducible subquotients of Verma modules. In the case at handthe statement is as follows. A weight µ = (µ, n) appears as the highest weightof an irreducible subquotient Mbλ,κc

, where λ = (λ, 0), if and only if n ≤ 0


and either µ = λ or there exists a finite sequence of weights µ0, . . . , µm ∈ h∗

such that µ0 = µ, µm = 0, µi+1 = µi ±miβi for some positive roots βi andpositive integers mi which satisfy

2(µi + ρ, βi) = mi(βi, βi) (6.3.13)

(here (·, ·) is the inner product on h∗ induced by an arbitrary non-degenerateinvariant inner product on g).

Now observe that the equations (6.3.13) coincide with the equations ap-pearing in the analysis of irreducible subquotients of the Verma modules overg of highest weights in the orbit of λ under the ρ-shifted action of the Weylgroup. In other words, a weight µ = (µ, n) appears in the decomposition ofM0,κc if and only n ≤ 0 and µ = w(ρ) − ρ for some element w of the Weylgroup of g. But for any w, the weight w(ρ) − ρ equals the sum of negativesimple roots of g. Hence the weight of any irreducible subquotient of M0,κc

has the form −nδ −∑

imiαi, n ≥ 0,mi ≥ 0. Such a weight cannot be of theform (6.3.12).

Therefore W+0,κc

is generated by the highest weight vector. Therefore thehomomorphism M0,κc

→ W+0,κc

is surjective. Since the characters of the twomodules coincide, we find that W+

0,κcis isomorphic to M0,κc

.

In Proposition 9.5.1 we will generalize this result to the case of Vermamodules of other highest weights.

In Section 8.1.1 we will need one more result on the structure of W+0,κc

.Consider the Lie algebra b+ = (b+ ⊗ 1)⊕ (g⊗ tC[[t]]).

Lemma 6.3.4 The space of b+-invariants of W+0,κc

is equal to π0 ⊂W+0,κc

.

Proof. It follows from the formulas for the action of gκc on W+0,κc

givenat the beginning of this section that all vectors in π0 are annihilated by b+.Let us show that there are no other b+-invariant vectors in W+

0,κc.

A b+-invariant vector is in particular annihilated by the Lie subalgebraL+n− = n− ⊗ tC[[t]]. In formula (6.1.3) we defined the operators eR

α,n, α ∈∆+, n ∈ Z. These operators generate the right action of the Lie algebra Ln+

on Mg, which commutes with the (left) action of Ln+ (which is part of the freefield realization of gκc). These operators now act on W+

0,κc, but because we

have applied the involution exchanging n+ and n−, we will now denote themby fR

α,n. They generate an action of the Lie algebra Ln−((t)) which commuteswith the action of Ln− which is part of the action of gκc on W+

0,κc(see the

formula at the beginning of this section).It is easy to see from the explicit formulas for these operators that the

lexicographically ordered monomials of the form∏la<0

bia,la

∏mb≤0

fRαb,mb

∏nc<0

a∗αc,nc|0〉′

6.4 Appendix: Proof of the Kac–Kazhdan conjecture 189

form a basis in W+0,κc

. Thus, we have a tensor product decomposition

W+0,κc

= W+

0,κc⊗W+,∗

0,κc,

where W+,∗0,κc

(resp., W+

0,κc) is the span of monomials in a∗α,n only (resp., in

fRα,m and bi,l only). Moreover, because the action of L+n− commutes withfR

α,m and bi,l, we find that L+n− acts only along the second factor of thistensor product decomposition.

Next, we prove, in the same way as in Lemma 5.6.4 that, as an L+n−-module, W+,∗

0,κcis isomorphic to the restricted dual of the free module with one

generator. Therefore the space of L+n−-invariants inW+,∗0,κc

is one-dimensional,spanned by the highest weight vector. We conclude that the space of L+n−-invariants of W+

0,κcis equal to the subspace W

+

0,κc.

Now suppose that we have a b+-invariant vector in W+0,κc

. Then it neces-

sarily belongs to W+

0,κc. But it is also annihilated by other elements of b+, in

particular, by h⊗ 1 ⊂ b+. Since

[h⊗ 1, a∗α,n] = α(h)a∗α,n, h ∈ h,

we find that a vector in W+

0,κcis annihilated by h ⊗ 1 if only if it belongs to

π0 (in which case it is annihilated by the entire Lie subalgebra b+). Hencethe space of b+-invariants of W+

0,κcis equal to π0.

6.4 Appendix: Proof of the Kac–Kazhdan conjecture

As an application of the Wakimoto modules, we give a proof of the Kac–Kazhdan conjecture from [KK], following [F1, F4].

We will use the extended affine Kac–Moody algebra, g′κ = CL0 n gκ and theweights of the extended Cartan subalgebra h = CL0⊕ (h⊗ 1), as described inthe proof of Proposition 6.3.3. It is known that the Verma module Mbλ,κ overgκ has a unique irreducible quotient, which we denote by Lbλ,κ.

Let us recall that in [KK] a certain subset Hκβ,m ∈ h∗ is defined for any

pair (β,m), where β is a positive root of gκ and m is a positive integer. Ifβ is a real root, then Hκ

β,m is a hyperplane in h∗ and if β is an imaginaryroot, then Hκc

β,m = h∗ and Hκβ,m = ∅ for κ 6= κc. It is shown in [KK] that

Lbλ,κ is a subquotient of Mbµ,κ if and only if the following condition is satisfied:

there exists a finite sequence of weights µ0, . . . , µn such that µ0 = λ, µn = µ,µi+1 = µi − miβi for some positive roots βi and positive integers mi, andµi ∈ Hκ

βi,mifor all i = 1, . . . , n.

Denote by ∆re+ the set of positive real roots of gκ (see the proof of Propo-

sition 6.3.3). Let us call a weight λ a generic weight of critical level ifλ does not belong to any of the hyperplanes Hκ

β,m, β ∈ ∆re+ . It is easy to


see from the above condition that λ is a generic weight of critical level if andonly if the only irreducible subquotients of Mbλ,κc

have highest weights λ−nδ,where n is a non-negative integer (i.e., their h∗ components are equal to theh∗ component of λ). The following assertion is the Kac–Kazhdan conjecturefor the untwisted affine Kac–Moody algebras.

Theorem 6.4.1 For generic weight λ of critical level

chLbλ,κc= e

bλ ∏α∈b∆re

+

(1− e−α)−1.

Proof. Without loss of generality, we may assume that λ = (λ, 0).Introduce the gradation operator L0 on the Wakimoto module Wχ(t) by us-

ing the vertex algebra gradation on Mg. It is clear from the formulas definingthe gκc

-action on Wχ(t) given in Theorem 6.1.6 that this action is compatiblewith the gradation if and only if χ(t) = λ/t, where λ ∈ h∗. In that case

chWλ/t = ebλ ∏

α∈b∆re+

(1− e−α)−1,

where λ = (λ, 0). Thus, in order to prove the theorem we need to show thatif λ is a generic weight of critical level, then Wλ/t is irreducible. Supposethat this is not so. Then either Wλ/t contains a singular vector, i.e., a vectorannihilated by the Lie subalgebra n+ = (g⊗ tC[[t]])⊕ (n+⊗1), other than themultiples of the highest weight vector, or Wλ/t is not generated by its highestweight vector.

Suppose that Wλ/t contains a singular vector other than a multiple of thehighest weight vector. Such a vector must then be annihilated by the Liesubalgebra L+n+ = n+[[t]]. We have introduced in Remark 6.1.1 the rightaction of n+((t)) on Mg, which commutes with the left action. It is clear fromformula (6.1.3) that the monomials∏

na<0

eRαa,na

∏mb≤0

a∗αb,mb|0〉 (6.4.1)

form a basis of Mg. We show in the same way as in the proof of Lemma 6.3.4that the space of L+n+-invariants of W0,κc

is equal to the subspace W 0,κcof

W0,κcspanned by all monomials (6.4.1) not containing a∗α,m’s.

In particular, we find that the weight of any singular vector of Wλ/t whichis not equal to the highest weight vector has the form (λ, 0)−

∑j(njδ − βj),

where nj > 0 and each βj is a positive root of g. But then Wλ/t contains anirreducible subquotient of such a weight.

Now observe that

ch M(λ,0),κc=∏n>0

(1− qn)−` · chWλ/t,

6.4 Appendix: Proof of the Kac–Kazhdan conjecture 191

where q = e−δ (see the proof of Proposition 6.3.3). If an irreducible moduleLbµ,κc

appears as a subquotient of Wλ/t, then it appears in the decompositionof chWλ/t into the sum of characters of irreducible representations and hencein the decomposition of ch M(λ,0),κc

. Since the characters of irreducible rep-resentations are linearly independent (see [KK]), this implies that Lbµ,κc

is anirreducible subquotient of M(λ,0),κc

. But this contradicts our assumption that(λ, 0) is a generic weight of critical level. Therefore we conclude that Wλ/t

does not contain any singular vectors other than the multiples of the highestweight vector.

Next, suppose that Wλ/t is not generated by its highest weight vector. Butthen there exists a homogeneous linear functional onWλ/t, whose weight is lessthan the highest weight and which is invariant under n− = (g⊗ t−1C[t−1])⊕(n−⊗ 1), and in particular, under its Lie subalgebra L−n+ = n+⊗ t−1C[t−1].Therefore this functional factors through the space of coinvariants of Wλ/t

by L−n+. But L−n+ acts freely on Wλ/t, and the space of coinvariants isisomorphic to the subspace C[a∗α,n]α∈∆+,n≤0 of Wλ/t. Hence we obtain thatthe weight of this functional has the form (λ, 0)−

∑j(njδ+βj), where nj ≥ 0

and each βj is a positive root of g. In the same way as above, it follows thatthis contradicts our assumption that λ is a generic weight. Therefore Wλ/t isgenerated by its highest weight vector. We also know that it does not containany singular vectors other than the highest weight vector. Hence Wλ/t isirreducible. This completes the proof.

7

Intertwining operators

We are now ready to prove Theorem 4.3.2. The proof is presented in thischapter and the next, following [F4]. The theorem was originally provedin [FF6, F1] (we note that a closely related statement, Theorem 8.3.1, wasconjectured by V. Drinfeld).

At the beginning of this chapter we outline the overall strategy of the proof(see Section 7.1) and then make the first two steps in the proof. We then de-velop the theory of intertwining operators between Wakimoto modules, whichis an important step of the proof of Theorem 4.3.2. First, we do it in Sec-tion 7.2 in the case of sl2. We construct explicitly intertwining operators,which we call the screening operators of the first and second kind. In Sec-tion 7.3 we use these operators and the functor of parabolic induction to con-struct intertwining operators between Wakimoto modules over an arbitraryaffine Kac–Moody algebra.

7.1 Strategy of the proof

Our strategy in proving Theorem 4.3.2 will be as follows. In Theorem 6.1.6we constructed a free field realization homomorphism of vertex algebras

wκc : Vκc(g)→Mg ⊗ π0. (7.1.1)

Step 1. We will show that the homomorphism (7.1.1) is injective.

Step 2. We will show that the image of z(g) ⊂ Vκc(g) under wκc is containedin π0 ⊂Mg ⊗ π0.

Thus, we need to describe the image of z(g) in π0.

Step 3. We will construct the screening operators Si, i = 1, . . . , `, fromW0,κc

= Mg ⊗ π0 to some other modules, which commute with the action ofgκc .

Step 4. We will show that the image of Vκc(g) under wκc

is contained in

192

7.1 Strategy of the proof 193⋂ì=1 KerSi. This implies that the image of z(g) is contained in

⋂ì=1 KerV i[1],

where V i[1] is the restriction of Si to π0.

Step 5. By using the associated graded of our modules and the isomorphismW+

0,κc' M0,κc from Proposition 6.3.3, we will find the character of z(g). We

will show that it is equal to the character of⋂`

i=1 KerV i[1]. Therefore we willobtain that

z(g) =⋂i=1

KerV i[1].

Step 6. By using Miura opers, we will show that there is a natural isomor-phism

FunOpLG(D) '⋂i=1

KerV i[1].

Therefore we obtain that

z(g) ' FunOpLG(D).

This will give us the proof of Theorem 6.1.6 because we will show that theabove identifications preserve the natural actions of the group AutO. We willalso show that this isomorphism satisfies various other compatibilities.

The proof of Theorem 6.1.6 presented in this book follows closely the paper[F4]. This proof is different from the original proof from [FF6, F1] in tworespects. First of all, we use the screening operators of the second kind ratherthan the first kind; their κ→ κc limits are easier to study. Second, we use theisomorphism between the Verma module of critical level with highest weight 0and a certain Wakimoto module and the computation of the associated gradedof the spaces of singular vectors to estimate the character of the center.

We now perform Steps 1 and 2 of the above plan. Then we will take upSteps 3 and 4 in the rest of this chapter.

7.1.1 Finite-dimensional case

We start with the statement of Step 1. In fact, we will prove a more generalresult that applies to an arbitrary κ, and not only to κc.

Before presenting the proof of this statement, it is instructive to considerits analogue in the finite-dimensional case.

Recall that in Section 5.2 we constructed the homomorphism (5.2.10),

ρ : U(g)→ Fun h∗ ⊗CD(N+). (7.1.2)

We wish to prove that this homomorphism is injective.

194 Intertwining operators

Consider the Poincare–Birkhoff–Witt filtration on U(g), and the filtrationon Fun h∗ ⊗

CD(N+) defined as follows: its nth term is the direct sum

n⊕i=0

(Fun h∗)≤i ⊗D(N+)≤(n−i).

Here (Fun h∗)≤i denotes the space of polynomials of degrees less than or equalto i, and D(N+)≤(n−i) is the space of differential operators of order less thanor equal to n− i.

It is clear from the formulas presented in Section 5.2 that the homomor-phism ρ preserves these filtrations. Consider the corresponding homomor-phism of the associated graded algebras. Recall that grU(g) = Fun g∗ andgrD(N+) = FunT ∗N+. Obviously, gr Fun h∗ = Fun h∗. Therefore we obtaina homomorphism of commutative algebras

Fun g∗ → Fun h∗ ⊗C

FunT ∗N+. (7.1.3)

It corresponds to a morphism of affine algebraic varieties

h∗ × T ∗N+ → g∗.

Observe that the cotangent bundle to N+ is identified with the trivial vectorbundle over N+ with the fiber n+. Since b+ = h⊕n, we find that h∗⊕n∗+ = b∗+.Hence the above morphism is equivalent to a morphism

b∗+ ×N+ → g∗.

By using a non-degenerate invariant inner product κ0 on g, we obtain a mor-phism

p : b− ×N+ → g. (7.1.4)

The latter morphism is easy to describe explicitly. It follows from the defini-tions that it sends

(x, g) ∈ b− ×N+ 7→ gxg−1 ∈ g.

Therefore the image of p in g consists of all elements of g which belong toa Borel subalgebra b such that the corresponding point of Fl = G/B− is inthe open N+-orbit U = N+ · [1] (we say that such b is in generic relativeposition with b−). Therefore the image of p is open and dense in g. In otherwords, p is dominant. Moreover, a generic element in the image is containedin a unique such Borel subalgebra b, so p is generically one-to-one. Thereforewe find that the homomorphism (7.1.3) is injective, which implies that thehomomorphism ρ of (7.1.2) is also injective.

Note that the morphism pmay be recast in the context of the Grothendieckalteration. Let g be the variety of pairs (b, x), where b is a Borel subalgebrain g and x ∈ b. There is a natural morphism g→ Fl, mapping (b, x) to b ∈ Fl.

7.1 Strategy of the proof 195

This map identifies g with a vector bundle over the flag variety Fl, whose fiberover b ∈ Fl is the vector space b. There is also a morphism g → g sending(b, x) to x. Now let U be the big cell, i.e., the open N+-orbit of Fl, and U itspreimage in g. In other words, U consists of those pairs (b, x) for which b is ingeneric relative position with b−. Then U is naturally isomorphic to b×N+

and the restriction of the morphism g→ g to U is the above morphism p.

7.1.2 Injectivity

Now we are ready to consider the analogous question for the affine Lie alge-bras, which corresponds to Step 1 of our plan.

Proposition 7.1.1 The homomorphism wκ of Theorem 6.2.1 is injective forany κ.

Proof. We apply the same argument as in the finite-dimensional case.Namely, we introduce filtrations on Vκ(g) and W0,κ = Mg ⊗ πκ−κc

0 which arepreserved by wκ, and then show that the induced map grwκ : grVκ(g) →grW0,κ (which turns out to be independent of κ) is injective.

The Poincare–Birkhoff–Witt filtration on U(gκ) induces a filtration on Vκ(g)as explained in Section 2.2.5.

Now we define a filtration W≤p0,κ on W0,κ By definition, W≤p

0,κ is the spanof monomials in the aα,n’s, a∗α,n’s and bi,n’s whose combined degree in theaα,n’s and bi,n’s is less than or equal to p (this is analogous to the filtrationused above in the finite-dimensional case). It is clear from the construction ofthe homomorphism wκ that it preserves these filtrations. Moreover, these arefiltrations of vertex algebras, and the associated graded spaces are commuta-tive vertex algebras, and in particular commutative algebras. The associatedgraded grwκ of wκ is therefore a homomorphism of these commutative alge-bras.

Now we describe the corresponding commutative algebras grVκ(g) andgrW0,κ and the homomorphism grwκ : grVκ(g) → grW0,κ. We will iden-tify g with g∗ using a non-degenerate invariant inner product κ0.

Let J(b−×N+) = Jb−×JN+ and Jg be the infinite jet schemes of b−×N+

and g, defined as in Section 3.4.2. Then grVκ(g) = FunJg and grW0,κ =FunJ(b− ×N+). The homomorphism

grwκ : FunJg→ FunJ(b− ×N+) (7.1.5)

corresponds to a morphism of jet schemes

J(b− ×N+)→ Jg.

It is clear from the construction that it is nothing but the morphism Jp

corresponding, by functoriality of J , to the morphism p given by formula(7.1.4).


But p is dominant and generically one-to one. Let ggen be the locus in theimage of p in g over which p is one-to-one. Then ggen is open and dense in g.Therefore Jggen is open and dense in Jg, and Jggen is clearly in the image ofJp. Hence we find that Jp is dominant and generically one-to-one. Thereforethe homomorphism of rings of functions (7.1.5) is injective. This implies thatwκ is also injective.

Now we specialize to the critical level. The vertex algebra Vκc(g) containsthe commutative subalgebra z(g), its center. Recall that z(g) is the spaceof g[[t]]-invariant vectors in Vκc

(g). On the other hand, W0,κc= Mg ⊗ π0

contains the commutative subalgebra π0, which is its center.

Lemma 7.1.2 The image of z(g) ⊂ Vκc(g) in W0,κc under wκc is containedin π0 ⊂W0,κc

.

Proof. We use the same argument as in the proof of Lemma 6.3.4.Let us observe that the lexicographically ordered monomials of the form∏

la<0

bia,la

∏mb<0

eRαb,mb

∏nc≤0

a∗αc,nc|0〉, (7.1.6)

where the eRα,n’s are given by formula (6.1.3), form a basis of W0,κc . Thus, we

have a tensor product decomposition

W0,κc= W 0,κc

⊗Mg,+,

where Mg,+ (resp., W 0,κc) is the span of monomials in a∗α,n only (resp., in

eRα,m and bi,l only).The image of any element of z(g) in W0,κc is an L+g-invariant vector, where,

as before, we use the notation L+g = g[[t]]. In particular, it is annihilated byL+n+ and by h. Since L+n+ commutes with eR

α,n and bi,n, it acts only alongthe second factor Mg,+ of the above tensor product decomposition. Accordingto Lemma 5.6.4, Mg,+ ' CoindL+g

L+b−C as an L+n+-module. Therefore the

space of L+n+-invariants in Mg,+ is one-dimensional, spanned by constants.Thus, we obtain that the space of L+n+-invariants in W0,κc

is equal toW 0,κc . However, the weight of the monomial (7.1.6) without the second factoris equal to the sum of positive roots corresponding to the factors eR

α,m. Sucha monomial is h-invariant if and only if there are no such factors present.Therefore the subspace of (h ⊕ L+n+)-invariants of W0,κc is the span of themonomials (7.1.6), which only involve the bi,l’s. This is precisely the subspaceπ0 ⊂W0,κc .

We are now done with Steps 1 and 2 of the proof of Theorem 4.3.2.

7.2 The case of sl2 197

7.2 The case of sl2

Now we perform Steps 3 and 4 of the plan outlined in Section 7.1. This willenable us to describe the image of z(g) inside π0 under wκc

as the intersectionof kernels of certain operators.

We start by giving a uniform construction of intertwining operators betweenWakimoto modules, the so-called screening operators (of two kinds) following[FF2, FF3, F1, FFR, F4]. First, we analyze in detail the case of g = sl2.

7.2.1 Vertex operators associated to a module over a vertex

algebra

We need to recall some general results on the vertex operators associatedto a module over a vertex algebra, following [FHL], Section 5.1. Let V bea conformal vertex algebra V (see the definition in Section 3.1.4) and M aV -module, i.e., a vector space together with a linear map

YM : V → EndM [[z±1]]

satisfying the axioms of Definition 5.1.1 of [FB]. In particular, the Fouriercoefficients of

YM (ω, z) =∑n∈Z

LMn z−n−2,

where ω is the conformal vector of V , define an action of the Virasoro algebraon M . We denote LM

−1 by T .Define a linear map

YV,M : M → Hom(V,M)[[z±1]]

by the formula

YV,M (A, z)B = ezTYM (B,−z)A, A ∈M,B ∈ V (7.2.1)

(compare with the skew-symmetry property of Proposition 2.3.2).This is an example of intertwining operators introduced in [FHL]. By

Proposition 5.1.2 of [FHL], this map satisfies the following property: for anyA ∈ V,B ∈M,C ∈ V , there exists an element

f ∈M [[z, w]][z−1, w−1, (z − w)−1]

such that the formal power series

YM (A, z)YV,M (B,w)C, YV,M (B,w)Y (A, z)C,

YV,M (YV,M (B,w − z)A, z)C, YV,M (Y (A, z − w)B,w)C

are expansions of f in

M((z))((w)), M((w))((z)), M((z))((z − w)), M((w))((z − w)),


respectively (compare with Corollary 3.2.3 of [FB]). Abusing notation, wewill write

YM (A, z)YV,M (B,w) = YV,M (Y (A, z − w)B,w),

and call this formula the operator product expansion (OPE), as in the caseM = V when YV,M = Y (see Section 2.3.3).

In the formulas below we will use the same notation Y (A, z) for Y (A, z)and YM (A, z). The following commutation relations between the Fourier co-efficients of Y (A, z) and YV,M (B,w) are proved in exactly the same way as inthe case M = V (see [FB], Section 3.3.6). If we write

Y (A, z) =∑n∈Z

A(n)z−n−1, YV,M (B,w) =

∑n∈Z

B(n)w−n−1,

then we have

[B(m), A(k)] =∑n≥0

(m

n

)(B(n)A)(m+k−n). (7.2.2)

In particular, we obtain that[∫YV,M (B, z)dz, Y (A,w)

]= YV,M

(∫YV,M (B, z)dz ·A,w

). (7.2.3)

Here, as before,∫

denotes the residue of at z = 0.Another property that we will need is the following analogue of formula

(2.3.5):

YV,M (TA, z) = ∂zYV,M (A, z). (7.2.4)

It is proved as follows:

YV,M (TA, z)B = ezTYM (B,−z)TA =

ezTTYM (B,−z)A− ezT [T, YM (B,−z)]A = ∂z(YV,M (A, z)B),

where we use the identity

[T, YM (B, z)] = ∂zYM (B, z),

which follows from [FB], Proposition 5.1.2.

7.2.2 The screening operator

Let us apply the results of Section 7.2.1 in the case of the vertex algebra W0,k

and its module W−2,k for k 6= −2. As before, we denote by |0〉 and | − 2〉 thehighest weight vectors of these modules.

Recall from Section 6.2.2 that under the homomorphism wk the generatorsen of sl2 are mapped to an (from now on, by abuse of notation, we will identify


the elements of sl2 with their images under wk). The commutation relationsof sl2 imply the following formulas:

[en, a−1] = 0, [hn, a−1] = 2an−1, [fn, a−1] = −hn−1 + kδn,1.

In addition, it follows from formulas (6.2.3) that

en| − 2〉 = an| − 2〉 = 0, n ≥ 0; hn| − 2〉 = fn| − 2〉 = 0, n > 0,

h0| − 2〉 = −2| − 2〉.

Therefore

en · a−1| − 2〉 = hn · a−1| − 2〉 = 0, n ≥ 0,

and

fn · a−1| − 2〉 = 0, n > 1.

We also find that

f1 · a−1| − 2〉 = (k + 2)| − 2〉,

and

f0 · a−1| − 2〉 = a−1f0| − 2〉 − h0| − 2〉= −b−1| − 2〉 = (k + 2)T | − 2〉, (7.2.5)

where T is the translation operator. The last equality follows from the formulafor the conformal vector in W0,k given in Proposition 6.2.2, which implies thatthe action of T on π0 ⊂W0,k is given by

T =1

2(k + 2)

∑n∈Z

bnb−n−1.

We wish to write down an explicit formula for the operator

Sk(z) def= YW0,k,W−2,k(a−1| − 2〉) : W0,k →W−2,k

and use the above properties to show that its residue is an intertwining oper-ator, i.e., it commutes with the action of sl2,k. Since the vertex subalgebrasMsl2 and πk+2

0 of W0,k commute with each other, we find that

Sk(z) = YMsl2(a−1|0〉, z)Yπk+2

0 ,πk+2−2

(| − 2〉, z) = a(z)Yπk+20 ,πk+2

−2(| − 2〉, z).

It remains to determine

V−2(z)def= Yπk+2

0 ,πk+2−2

(| − 2〉, z) : πk+20 → πk+2

−2 .

The identity (7.2.2) specialized to the case A = b−1|0〉 and B = | − 2〉implies the following commutation relations:

[bn, V−2(z)] = −2znV−2(z). (7.2.6)


In addition, we obtain from formulas (7.2.4) and (7.2.5) that

(k + 2)∂zV−2(z) = −:b(z)V−2(z): . (7.2.7)

It is easy to see that formulas (7.2.6) and (7.2.7) determine V−2(z) uniquely(this is explained in detail in [FB], Section 5.2.6). As the result, we obtainthe following explicit formula:

V−2(z) = T−2 exp

(1

k + 2

∑n<0

bnnz−n

)exp

(1

k + 2

∑n>0

bnnz−n

). (7.2.8)

Here we denote by T−2 the translation operator πk+20 → πk+2

−2 sending thehighest weight vector |0〉 to the highest weight vector | − 2〉 and commutingwith all bn, n 6= 0.

Now, using formula (7.2.2) we obtain that the operator Sk(z) has the fol-lowing OPEs:

e(z)Sk(w) = reg., h(z)Sk(w) = reg.,

f(z)Sk(w) =(k + 2)V−2(w)

(z − w)2+

(k + 2)∂wV−2(w)z − w

+ reg.

= (k + 2)∂wV−2(w)z − w

+ reg.

Since the residue of a total derivative is equal to 0, this implies the following:

Proposition 7.2.1 The residue Sk =∫Sk(w)dw is an intertwining operator

between the sl2-modules W0,k and W−2,k.

We call Sk the screening operator of the first kind for sl2.

Proposition 7.2.2 For k 6∈ −2 + Q≥0 the sequence

0→ Vk(sl2)→W0,kSk→W−2,k → 0

is exact.

Proof. By Proposition 7.1.1, Vk(sl2) is naturally an sl2-submodule of W0,k

for any value of k. The module Vk(sl2) is generated from the vacuum vector|0〉, whose image in W0,k is the highest weight vector. We have

Sk =∑n∈Z

anV−2,−n,

where V−2,−n is the coefficient in front of zn in V−2(z). It is clear from formula(7.2.8) that V−2,m|0〉 = 0 for all m > 0. We also have an|0〉 = 0 for all n ≥ 0.Therefore |0〉 belongs to the kernel of Sk. But since Sk commutes with theaction of sl2, this implies that the entire submodule Vk(sl2) lies in the kernelof Sk.


In order to prove that Vk(sl2) coincides with the kernel of Sk, we comparetheir characters (see the proof of Proposition 6.3.3 for the definition of thecharacters). We will use the notation q = e−δ, u = eα.

Since Vk(sl2) is isomorphic to the universal enveloping algebra of the Liealgebra spanned by en, hn and fn with n < 0, we find that

chVk(g) =∏n>0

(1− qn)−1(1− uqn)−1(1− u−1qn)−1. (7.2.9)

Similarly, we obtain that

chWλ,k = uλ/2∏n>0

(1− qn)−1(1− uqn)−1(1− u−1qn−1)−1.

Thus, chWλ,k = ch Mλ,k, where Mλ,k is the Verma module over sl2 withhighest weight (λ, k). Therefore if Mλ,k is irreducible, then so is Wλ,k. Theset of values (λ, k) for which Mλ,k is irreducible is described in [KK]. It followsfrom this description that if k 6∈ −2 + Q≥0, then M−2,k, and hence W−2,k, isirreducible. It is easy to check that Sk(a∗0|0〉) = |−2〉, so that Sk is a non-zerohomomorphism. Hence it is surjective for such values of k. Therefore thecharacter of its kernel for such k is equal to chW0,k − chW−2,k = chVk(sl2).This completes the proof.

Next, we will describe the second screening operator for sl2. For this weneed to recall the Friedan–Martinec–Shenker bosonization.

7.2.3 Friedan–Martinec–Shenker bosonization

Consider the Heisenberg Lie algebra with the generators pn, qn, n ∈ Z, andthe central element 1 with the commutation relations

[pn, pm] = nδn,−m1, [qn, qm] = −nδn,−m1, [pn, qm] = 0.

We set

p(z) =∑n∈Z

pnz−n−1, q(z) =

∑n∈Z

qnz−n−1.

For λ ∈ C, µ ∈ C, let Πλ,µ be the Fock representation of this Lie algebragenerated by a highest weight vector |λ, µ〉 such that

pn|λ, µ〉 = λδn,0|λ, µ〉, qn|λ, µ〉 = µδn,0|λ, µ〉, n ≥ 0; 1|λ, µ〉 = |λ, µ〉.

Consider the vertex operators Vλ,µ(z) : Πλ′,µ′ → Πλ+λ′,µ+µ′ given by theformula

Vλ,µ(z) =

Tλ,µzλλ′−µµ′ exp

(−∑n<0

λpn + µqnn

z−n

)exp

(−∑n>0

λpn + µqnn

z−n

),


where Tλ,µ is the translation operator Π0,0 → Πλ,µ sending the highest weightvector to the highest weight vector and commuting with all pn, qn, n 6= 0.

Abusing notation, we will write these operators as eλu+µv, where u(z) andv(z) stand for the anti-derivatives of p(z) and q(z), respectively, i.e., p(z) =∂zu(z), q(z) = ∂zv(z).

For γ ∈ C, set

Πγ =⊕n∈Z

Πn+γ,n+γ .

Using the vertex operators, one defines a vertex algebra structure on the directsum Π0 (with the vacuum vector |0, 0〉) as in [FB], Section 5.2.6.

Moreover, Πγ is a module over the vertex algebra Π0 for any γ ∈ C.The following realization of the vertex algebra M (also known as the “βγ-

system”) in terms of the vertex algebra Π0 is due to Friedan, Martinec, andShenker [FMS]. (The isomorphism between the image of M in Π0 and the

kernel of∫eudz stated in the theorem was established in [FF5].)

Theorem 7.2.3 There is a (unique) embedding of vertex algebras M → Π0

under which the fields a(z) and a∗(z) are mapped to the fields

a(z) = eu+v, a∗(z) = (∂ze−u)e−v = −:p(z)e−u−v: .

Further, the image of M in Π0 is equal to the kernel of the operator∫eudz.

Equivalently, Π0 may be described as the localization of M with respect toa−1, i.e.,

Π0 'M [(a−1)−1] = C[an]n<−1 ⊗ C[a∗n]n≤0 ⊗ C[(a−1)±1]. (7.2.10)

The vertex algebra structure on Π0 is obtained by a natural extension of thevertex algebra structure on M .

Under the embedding M → Π0 the Virasoro field T (z) = :∂za∗(z)a(z): of

M described in Section 6.2.3 is mapped to the following field in Π0:

12:p(z)2:− 1

2∂zp(z)−

12:q(z)2: +

12∂zq(z).

Thus, the map M → Π0 becomes a homomorphism of conformal vertex alge-bras with respect to the conformal structures corresponding to these fields.

The reason why the FMS bosonization is useful to us is that it allows usto make sense of the field a(z)γ , where γ is an arbitrary complex number.Namely, we replace a(z)γ with the field

a(z)γ = eγ(u+v) : Π0 → Πγ ,

which is well-defined.


Now we take the tensor product Π0 ⊗ πk+20 , where we again assume that

k 6= −2. This is a vertex algebra which contains M⊗πk+20 , and hence Vk(sl2),

as vertex subalgebras. In particular, for any γ, λ ∈ C, the tensor productΠγ ⊗ πk+2

λ is a module over Vk(sl2) and hence over sl2. We denote it byWγ,λ,k. In addition, we introduce the bosonic vertex operator

V2(k+2)(z) = T2(k+2) exp

(−∑n<0

bnnz−n

)exp

(−∑n>0

bnnz−n

). (7.2.11)

Let us set

Sk(z) = a(z)−(k+2)V2(k+2)(z). (7.2.12)

A straightforward computation similar to the one performed in Section 7.2.2yields:

Proposition 7.2.4 The residue

Sk =∫Sk(z)dz : W0,0,k → W−(k+2),2(k+2),k

is an intertwining operator between the sl2-modules W0,0,k and W−(k+2),2(k+2),k.

We call Sk, or its restriction to W0,k ⊂ W0,0,k, the screening operator ofthe second kind for sl2. This operator was first introduced by V. Dotsenko[Do].

Proposition 7.2.5 For generic k the sl2-submodule Vκ(sl2) ⊂ W0,k is equalto the kernel of Sk : W0,k → W−(k+2),2(k+2),k.

Proof. We will show that for generic k the kernel of Sk : W0,k →W−(k+2),2(k+2),k coincides with the kernel of the screening operator of thefirst kind, Sk : W0,k → W−2,k. This, together with Proposition 7.2.2, willimply the statement of the proposition.

Note that the operator Sk has an obvious extension to an operator W0,0,k →W0,−2,k defined by the same formula. The kernel of Sk (resp., Sk) in W0,k

is equal to the intersection of W0,k ⊂ W0,0,k and the kernel of Sk (resp.,Sk) acting from W0,0,k to W−(k+2),2(k+2),k (resp., W0,−2,k). Therefore it issufficient to show that the kernels of Sk and Sk on W0,0,k are equal for generick.

Now let φ(z) be the anti-derivative of b(z), i.e., b(z) = ∂zφ(z). By abusingnotation we will write V−2(z) = e−(k+2)−1φ and V2(k+2)(z) = eφ. Then thescreening currents Sk(z) and Sk(z) become

Sk(z) = eu+v−(k+2)−1φ(z), Sk(z) = e−(k+2)u−(k+2)v+φ.

Consider a more general situation: let h be an abelian Lie algebra with anon-degenerate inner product κ. Using this inner product, we identify h with


h∗. Let hκ be the Heisenberg Lie algebra and πκλ, λ ∈ h∗ ' h be its Fock

representations, defined as in Section 6.2.1. Then for any χ ∈ h∗ there is avertex operator V κ

χ (z) : πκ0 → πκ

χ given by the formula

V κχ (z) = Tχ exp

(−∑n<0

χn

nz−n

)exp

(−∑n>0

χn

nz−n

), (7.2.13)

where the χn = χ ⊗ tn are the elements of the Heisenberg Lie algebra hκ

corresponding to χ ∈ h∗, which we identify with h using the inner product κ.Suppose that κ(χ, χ) 6= 0, and denote by χ the element of h equal to

−2χ/κ(χ, χ). We claim that if χ is generic (i.e., away from countably many

hypersurfaces in h), then the kernels of∫V κ

χ (z)dz and∫V κ

χ (z)dz in πκ0

coincide. Indeed, we have a decomposition of hκ into a direct sum of theHeisenberg Lie subalgebra hχ

κ generated by χn, n ∈ Z, and the HeisenbergLie subalgebra h⊥κ which is the centralizer of χn, n ∈ Z, in hκ (this is aHeisenberg Lie subalgebra of hκ corresponding to the orthogonal complementof χ in h; note that by our assumption on χ, this orthogonal complementdoes not contain χ). The Fock representation πκ

0 decomposes into a tensorproduct of Fock representations of these two Lie subalgebras. The operators∫V κ

χ (z)dz and∫V κ

χ (z)dz commute with h⊥κ . Therefore the kernel of each of

these operators in πκ0 is equal to the tensor product of the Fock representation

of h⊥κ and the kernel of this operator on the Fock representation of hχκ.

In other words, the kernel is determined by the corresponding kernel on theFock representation of the Heisenberg Lie algebra hχ

κ. The latter kernels for∫V κ

χ (z)dz and∫V κ

χ (z)dz coincide for generic values of κ(χ, χ), as shown in

[FB], Section 15.4.15. Hence the kernels of∫V κ

χ (z)dz and∫V κ

χ (z)dz also

coincide generically.Now we apply this result in our situation, which corresponds to the three-

dimensional Lie algebra h with a basis u, v, φ and the following non-zero innerproducts of the basis elements:

κ(u, u) = −κ(v, v) = 1, κ(φ, φ) = 2(k + 2).

Our screening currents Sk(z) and Sk(z) are equal to V κχ (z) and V κ

χ (z), where

χ = u+ v − (k + 2)−1φ, χ = −(k + 2)χ = −(k + 2)u− (k + 2)v + φ.

Therefore for generic k the kernels of the screening operators Sk and Sk co-incide.

7.3 Screening operators for an arbitrary g 205

7.3 Screening operators for an arbitrary g

In this section we construct screening operators between Wakimoto modulesover gκ for an arbitrary simple Lie algebra g and use them to characterizeVκ(g) inside W0,κ.

7.3.1 Parabolic induction

Denote by sl(i)2 the Lie subalgebra of g, isomorphic to sl2, which is generated

by ei, hi, and fi. Let p(i) be the parabolic subalgebra of g spanned by b− andei, and m(i) its Levi subalgebra. Thus, m(i) is equal to the direct sum of sl

(i)2

and the orthogonal complement h⊥i of hi in h.We apply to p(i) the results on semi-infinite parabolic induction of Sec-

tion 6.3. According to Corollary 6.3.2, we obtain a functor from the categoryof smooth representations of sl2,k ⊕ h⊥i,κ0

, with k and κ0 satisfying the con-ditions of Theorem 6.3.1, to the category of smooth gκ+κc

-modules. Thecondition on k and κ0 is that the inner products on sl

(i)2 corresponding to

(k + 2) and κ0 are both restrictions of an invariant inner product (κ − κc)on g. In other words, (κ − κc)(hi, hi) = 2(k + 2) and κ0 = κ|h⊥i . By abuseof notation we will write κ for κ0. If k and κ satisfy this condition, then forany smooth sl2-module R of level k and any smooth h⊥κ -module L the tensorproduct Mg,p(i) ⊗R⊗ L is a smooth gκ+κc -module.

Note that the above condition means that

(κ− κc)(hi, hj) = βaji, (7.3.1)

where aji is the (ji)th entry of the Cartan matrix of g.In particular, if we choose R to be the Wakimoto module Wλ,k over sl2,

and L to be the Fock representation πκλ0

, the corresponding gκ-module willbe isomorphic to the Wakimoto module W(λ,λ0),κ+κc

, where (λ, λ0) is theweight of g built from λ and λ0. Under this isomorphism the generatorsaαi,n, n ∈ Z, will have a special meaning: they correspond to the right actionof the elements ei,n of g, which was defined in Section 6.1.1. In other words,making the above identification of modules forces us to choose a system ofcoordinates yαα∈∆+ on N+ such that ρR(ei) = ∂/∂yαi (in the notation ofSection 5.2.5), and so wR(ei(z)) = aαi

(z) (in the notation of Section 6.1.1).From now on we will denote wR(ei(z)) by eR

i (z). For a general coordinatesystem on N+ we have

eRi (z) = aαi(z) +

∑β∈∆+

PR,iβ (a∗α(z))aβ(z) (7.3.2)

(see formula (6.1.1)).Now any intertwining operator between Wakimoto modulesWλ1,k andWλ2,k

over sl2 gives rise to an intertwining operator between the Wakimoto modulesW(λ1,λ0),κ+κc

and W(λ2,λ0),κ+κcover gκ+κc

for any weight λ0 of h⊥i . We will


use this fact and the sl2 screening operators introduced in the previous sectionto construct intertwining operators between Wakimoto modules over gκ.

7.3.2 Screening operators of the first kind

Let κ be a non-zero invariant inner product on g. We will use the samenotation for the restriction of κ to h. Now to any χ ∈ h we associate a vertexoperator V κ

χ (z) : πκ0 → πκ

χ given by formula (7.2.13). For κ 6= κc, we set

Si,κ(z) def= eRi (z)V κ−κc

−αi(z) : W0,κ →W−αi,κ,

where eRi (z) is given by formula (6.1.1). Note that

Si,κ(z) = YW0,κ,W−αi,κ(eRi,−1| − αi〉, z) (7.3.3)

in the notation of Section 7.2.1.According to Proposition 7.2.1 and the above discussion, the operator

Si,κ =∫Si,κ(z)dz : W0,κ →W−αi,κ (7.3.4)

is induced by the screening operator of the first kind Sk for the ith sl2 sub-algebra, where k is determined from the formula (κ − κc)(hi, hi) = 2(k + 2).Hence Proposition 7.2.1 implies:

Proposition 7.3.1 The operator Si,κ is an intertwining operator between thegκ-modules W0,κ and W−αi,κ for each i = 1, . . . , `.

We call Si,κ the ith screening operator of the first kind for gκ.Recall that by Proposition 7.1.1 Vκ(g) is naturally a gκ-submodule and a

vertex subalgebra of W0,κ. On the other hand, the intersection of the kernelsof Si,κ, i = 1, . . . , `, is a gκ-submodule of W0,κ, by Proposition 7.3.1, anda vertex subalgebra of W0,κ, due to formula (7.3.3) and the commutationrelations (7.2.3). The following proposition is proved in [FF8] (we will not useit here).

Proposition 7.3.2 For generic κ, Vκ(g) is equal to the intersection of thekernels of the screening operators Si,κ, i = 1, . . . , `.

Furthermore, in [FF8], Section 3, a complex C•κ(g) of gκ-modules is con-structed for generic κ. Its ith degree term is the direct sum of the Wakimotomodules Ww(ρ)−ρ,κ, where w runs over all elements of the Weyl group of g oflength i. Its zeroth cohomology is isomorphic to Vκ(g), and all other coho-mologies vanish. For g = sl2 the complex C•κ(sl2) has length 1 and coincideswith the one appearing in Proposition 7.2.2. In general, the degree 0 termC0

κ(g) of the complex is W0,κ, the degree 1 term C1κ(g) is

⊕ì=1W−αi,κ. The

zeroth differential is the sum of the screening operators Si,κ.


In [FF8] it is also explained how to construct other intertwining operatorsas compositions of the screening operators Si,κ using the Bernstein–Gelfand–Gelfand resolution of the trivial representation of the quantum group Uq(g).Roughly speaking, the screening operators Si,κ satisfy the q-Serre relations,i.e., the defining relations of the quantized enveloping algebra Uq(n+) withappropriate parameter q. Then for generic κ we attach to a singular vectorof weight µ in the Verma module Mλ over Uq(g) an intertwining operatorWλ,κ → Wµ,κ. This operator is equal to the integral of a product of thescreening currents Si,κ(z) over a certain cycle on the configuration space withcoefficients in a local system that is naturally attached to the above singularvector.

The simplest operators correspond to the singular vectors fiv0 of weight−αi in the Verma module M0 over Uq(g). The corresponding intertwiningoperators are nothing but our screening operators (7.3.4).

7.3.3 Screening operators of the second kind

In order to define the screening operators of the second kind, we need to makesense of the series (eR

i (z))γ for complex values of γ. So we choose a system ofcoordinates on N+ in such a way that eR

i (z) = aαi(z) (note that this cannotbe achieved for all i = 1, . . . , ` simultaneously). This is automatically so if wedefine the Wakimoto modules over g via the semi-infinite parabolic inductionfrom Wakimoto modules over the ith subalgebra sl2 (see Section 7.3.1).

Having chosen such a coordinate system, we define the series aαi(z)γ using

the Friedan–Martinec–Shenker bosonization of the Weyl algebra generated byaαi,n, a

∗αi,n, n ∈ Z, as explained in Section 7.2.3. Namely, we have a vertex

algebra

Π(i)0 = C[aαi,n]n<−1 ⊗ C[a∗αi,n]n≤0 ⊗ C[a±1

αi,−1]

containing

M(i)g = C[aαi,n]n≤−1 ⊗ C[a∗αi,n]n≤0

and a Π(i)0 -module Π(i)

γ defined as in Section 7.2.3. We then set

W(i)γ,λ,κ = Wλ,κ ⊗

M(i)g

Π(i)γ .

This is a gκ-module, which contains Wλ,κ if γ = 0. Note that W (i)0,0,κc

is the g-module obtained by the semi-infinite parabolic induction from the sl2-moduleW0,0,−2.

Now let β = 12 (κ− κc)(hi, hi) and define the field

Sκ,i(z)def= (eR

i (z))−βVαi(z) : W (i)0,0,κ → W

(i)−β,βαi,κ

. (7.3.5)


Here αi = hi ∈ h denotes the ith coroot of g. Then the operator

Sκ,i =∫Sκ,i(z)dz

is induced by the screening operator of the second kind Sk for the ith sl2subalgebra, where k is determined from the formula (κ−κc)(hi, hi) = 2(k+2).Hence Proposition 7.2.1 implies (a similar result was also obtained in [PRY]):

Proposition 7.3.3 The operator Sκ,i is an intertwining operator between thegκ-modules W (i)

0,0,κ and W (i)−β,βαi,κ

.

Note that Si,κ is the residue of YV,M ((eRi,−1)

−β |αi〉, z), where V = W(i)0,0,κ

and M = W(i)−β,βαi,κ

(see Section 7.2.1). Therefore, according to the com-mutation relations (7.2.3), the intersection of kernels of Si,κ, i = 1, . . . , `, isnaturally a vertex subalgebra of W (i)

0,0,κ or W0,κ. Combining Proposition 7.3.2and Proposition 7.2.5, we obtain:

Proposition 7.3.4 For generic κ, Vκ(g) is isomorphic, as a gκ-module andas a vertex algebra, to the intersection of the kernels of the screening operators

Si,κ : W0,κ → W(i)−β,βαi,κ

, i = 1, . . . , `.

We remark that one can use the screening operators Si,κ to construct moregeneral intertwining operators following the procedure of [FF8].

7.3.4 Screening operators of second kind at the critical level

We would like to use the screening operators Si,κ to characterize the imageof the homomorphism Vκc

(g)→W0,κc(implementing Step 3 of our plan from

Section 7.1). First, we need to define their limits as κ→ κc.We start with the case when g = sl2. In order to define the limit of Sk

as k → −2 we make W0,k and W−(k+2),2(k+2),k into free modules over C[β],where β is a formal variable representing k+2, and then consider the quotientof these modules by the ideal generated by β.

More precisely, let π0[β] (resp., π2β [β]) be the free C[β]-module spanned bythe monomials in bn, n < 0, applied to a vector |0〉 (resp., |2β〉). We define thestructure of vertex algebra over C[β] on π0[β] as in Section 6.2.1. Then π2β [β]is a module over π0[β]. The dependence on β comes from the commutationrelations

[bn, bm] = 2βnδn,−m

and the fact that b0 acts on π2β [β] by multiplication by 2β. Taking thequotient of π0[β] (resp., π2β [β]) by the ideal generated by (β − k), k ∈ C, we


obtain the vertex algebra πk0 (resp., the module πk

2k over πk0 ) introduced in

Section 6.2.2.We define free C[β]-modules W0[β] and W0,0[β] as the tensor products

M ⊗C π0[β] and Π0 ⊗C π0[β],

respectively. These are vertex algebras over C[β], and their quotients by theideals generated by (β − k), k ∈ C, are the vertex algebras W0,k and W0,0,k,respectively.

Next, let Π−β+n,−β+n be the free C[β]-module spanned by the lexicograph-ically ordered monomials in pn, qn, n < 0, applied to a vector |−β+n,−β+n〉.Set

Π−β =⊕n∈Z

Π−β+n,−β+n,

and

W−β,2β = Π−β ⊗C[β] π2β [β].

Each Fourier coefficient of the formal power series

V2β(z) : π0[β]→ π2β [β]⊗C C[β], a(z)−β : Π0 ⊗C C[β]→ Π−β ,

given by the formulas above, is a well-defined linear operator commuting withthe action of C[β]. Hence the Fourier coefficients of their product are also well-defined linear operators from W0[β] to W−β,2β . The corresponding operatorson the quotients by the ideals generated by (β − k), k ∈ C, coincide withthe operators introduced above. We need to compute explicitly the leadingterm in the β-expansion of the residue

∫a(z)−βV2β(z)dz and its restriction to

W0[β]. We will use this leading term as the screening operator at the criticallevel.

Consider first the expansion of V2β(z) in powers of β. Let us write V2β(z) =∑n∈Z V2β [n]z−n. Introduce the operators V [n], n ≤ 0, via the formal power

series ∑n≤0

V [n]z−n = exp

(∑m>0

b−m

mzm

). (7.3.6)

Using formula (7.2.11), we obtain the following expansion of V2β [n]:

V2β [n] =

V [n] + β(. . .), n ≤ 0,

βV [n] + β2(. . .), n > 0,

where

V [n] = −2∑m≤0

V [m]∂

∂bm−n, n > 0. (7.3.7)

Next, we consider the expansion of a(z)−β = e−β(u+v) in powers of β. Let


us write a(z)−β =∑

n∈Z a(z)−β[n] z

−n. We will identify Π−β with Π0 ⊗C C[β],as C[β]-modules. Then we find that

a(z)−β[n] =

1 + β(. . .), n = 0,

βpn + qn

n+ β2(. . .), n 6= 0.

The above formulas imply the following expansion of the screening operatorSβ−2:∫

a(z)−βV2β(z)dz = β

(V [1] +

∑n>0

1nV [−n+ 1](pn + qn)

)+ β2(. . .).

Therefore we define the limit of the screening operator Sβ−2 at β = 0 (corre-sponding to k = −2) as the operator

Sdef= V [1] +

∑n>0

1nV [−n+ 1](pn + qn), (7.3.8)

acting from W0,−2 = Msl2 ⊗ π0 to W0,0,−2 = Π0 ⊗ π0. By construction, S isequal to the leading term in the β-expansion of the screening operator Sβ−2.Hence S commutes with the sl2-action on W0,−2 and W0,0,−2.

It is possible to express the operators

pn + qnn

= −(un + vn), n > 0,

in terms of the Heisenberg algebra generated by am, a∗m,m ∈ Z. Namely, from

the definition a(z) = eu+v it follows that u(z) + v(z) = log a(z), so that thefield u(z) + v(z) commutes with a(w), and we have the following OPE witha∗(w):

(u(z) + v(z))a∗(w) =1

z − wa(w)−1 + reg.

Therefore, writing a(z)−1 =∑

n∈Z a(z)−1[n]z−n, we obtain the following com-

mutation relations:[pn + qn

n, a∗m

]= −a(z)−1

[n+m−1], n > 0. (7.3.9)

Using the realization (7.2.10) of Π0, the series a(z)−1 is expressed as follows:

a(z)−1 = (a−1)−1

1 + (a−1)−1∑

n 6=−1

anz−n−1

−1

, (7.3.10)

where the right hand side is expanded as a formal power series in positivepowers of (a−1)−1. It is easy to see that each Fourier coefficient of this powerseries is well-defined as a linear operator on Π0.

The above formulas completely determine the action of (pn + qn)/n, n > 0,


and hence of S, on any vector in W0,−2 = M ⊗π0 ⊂ W0,0,−2. Namely, we usethe commutation relations (7.3.9) to move (pn + qn)/n through the a∗m’s. Asthe result, we obtain Fourier coefficients of a(z)−1, which are given by formula(7.3.10). Applying each of them to any vector in W0,−2, we always obtain afinite sum.

This completes the construction of the limit S of the screening operator Sk

as k → −2 in the case of sl2. Now we consider the case of an arbitrary g.The limit of Si,κ as κ→ κc is by definition the operator

Si : W0,κc→ W

(i)0,0,κc

,

obtained from S via the functor of semi-infinite parabolic induction. Thereforeit is given by the formula

Si = V i[1] +∑n>0

1nV i[−n+ 1](pi,n + qi,n). (7.3.11)

Here V i[n] : π0 → π0 are the linear operators given by the formulas∑n≤0

V i[n]z−n = exp

(∑m>0

bi,−m

mzm

), (7.3.12)

V i[1] = −∑m≤0

V i[m] Dbi,m−1 , (7.3.13)

where Dbi,mdenotes the derivative in the direction of bi,m given by the formula

Dbi,m· bj,n = ajiδn,m, (7.3.14)

and (akl) is the Cartan matrix of g (it is normalized so that we have Dbi,m ·bi,n = 2δn,m, as in the case of sl2). This follows from the commutationrelations (6.2.1) between the bi,n’s and formula (7.3.1).

The operators (pi,n + qi,n)/n acting on Π(i)0 are defined in the same way as

above.Thus, we obtain well-defined linear operators Si : W0,κc → W

(i)0,0,κc

. Byconstruction, they commute with the action of gκc

on both modules. It isclear that the operators Si, i = 1, . . . , `, annihilate the highest weight vectorof W0,κc

. Therefore they annihilate all vectors obtained from the highestweight vector under the action of gκc , i.e., all vectors in Vκc(g) ⊂W0,κc . Thuswe obtain

Proposition 7.3.5 The image of the vacuum module Vκc(g) under wκc

iscontained in the intersection of the kernels of the operators Si : W0,κc

→W

(i)0,0,κc

, i = 1, . . . , `.

It follows from Proposition 7.3.5, Proposition 7.1.1 and Lemma 7.1.2 thatthe image of z(g) under the homomorphism wκc : Vκc(g) →W0,κc is contained


in the intersection of the kernels of the operators Si, i = 1, . . . , `, restricted toπ0 ⊂ W0,κc

. But according to formula (7.3.11), the restriction of Si to π0 isnothing but the operator V i[1] : π0 → π0 given by formula (7.3.13). Thereforewe obtain the following:

Proposition 7.3.6 The center z(g) of Vκc(g) is contained in the intersectionof the kernels of the operators V i[1], i = 1, . . . , `, in π0.

This completes Step 4 of the plan outlined in Section 7.1. In the nextchapter we will use this result to describe the center z(g) and identify it withthe algebra of functions on the space of opers.

7.3.5 Other Fourier components of the screening curents

In Section 9.6 below we will need other Fourier components of the screen-ing currents of the second kind constructed above (analogous results may beobtained for the screening currents of the first kind along the same lines).

We start with the case of g = sl2. Consider the screening current

Sk(z) = a(z)−(k+2)V2(k+2)(z),

introduced in formula (7.2.12). In Proposition 7.2.4 we considered its residueas a linear operator W0,0,k → W−(k+2),2(k+2),k. In particular V2(k+2)(z) wasdefined in formula (7.2.11) as a current acting from π

(k+2)0 to πk+2

2(k+2).We may also consider an action of this current on other Fock modules

πk+2λ , λ ∈ C, over the Heisenberg algebra. However, in that case we also need

to multiply the current given by formula (7.2.11) by the factor zb0 = zλ. Onlythen will the relations between this current and b(z) obtained above be valid.(If λ = 0, then zλ = 1, and that is why we did not include it in formula(7.2.11)).) From now on we will denote by V2(k+2)(z) the current (7.2.11)multiplied by zb0 = zλ. This is a formal linear combination of zα, whereα ∈ λ + Z. Therefore for non-integer values of λ we cannot take its residue.But if λ ∈ Z, then we obtain a well-defined linear operator

S(λ)k

def=∫Sk(z)dz : Wλ,k → W−(k+2),λ+2(k+2),k, (7.3.15)

which again commutes with the action of sl2.Next, we consider the limit of this operator when k = β − 2 and β → 0.

Suppose that λ ∈ Z+. Then, in the same way as in Section 7.3.4, we find thatS

(λ)β−2 → βS

(λ), where

S(λ)

= V [λ+ 1] +∑n>0

1nV [−n+ 1](pn+λ + qn+λ), (7.3.16)

acting from Wλ,−2 = Msl2 ⊗ πλ to W0,λ,−2 = Π0 ⊗ πλ. The operators V [n]


acting on πλ are defined by formulas (7.3.6) and (7.3.7), and the operatorspn + qn acting from Msl2 to Π0 are defined by formulas (7.3.9) and (7.3.10).

Since S(λ)

is obtained as a limit of an intertwining operator, we obtain thatS

(λ)is also an intertwining operator (i.e., it commutes with the action of sl2).

Now we construct intertwining operators for an arbitrary g by applying thesemi-infinite parabolic induction to the operators (7.3.15) and (7.3.16), as inSection 7.3.4. First, for any weight λ ∈ h∗ such that λi = 〈λ, αi〉 ∈ Z weobtain the intertwining operator

S(λi)κ,i =

∫Sκ,i(z)dz : W

(i)λ,κ → W

(i)−β,λ+βαi,κ

,

where Sκ,i(z) is the current given by formula (7.3.5) multiplied by the factorzλi .

Next, we consider the κ→ κc limit of this operator for λi ∈ Z+. As before,this corresponds to the limit β → 0, where β = 1

2 (κ−κc)(hi, hi). The limitingoperator is given by the formula

S(λi)

i = V i[λi + 1] +∑n>0

1nV i[−n+ 1](pi,n+λi + qi,n+λi), (7.3.17)

where V i[n], n ≤ 0, are given by formula (7.3.12), and

V i[λi + 1] = −∑

m≤λi

V i[m− λi] Dbi,m−1 , (7.3.18)

where, as before, Dbi,mis the derivation on πλ ' C[bi,n]i=1,...,`;n<0 such that

Dbi,m · bj,n = ajiδn,m.

The operator S(λi)

i acts from Wλ,κc to W0,λ,κc . In particular, we find thatthe restriction of Si to πλ ⊂ Wλ,κc

is equal to V i[λi + 1], given by formula(7.3.18).

We will use this result in the proof of Proposition 9.6.7 below, which gen-eralizes Proposition 7.3.6 to arbitrary dominant integral weights.

8

Identification of the center with functions onopers

In this chapter we use the above description of the center of the vertex algebraVκc

(g) in terms of the kernels of the screening operators in order to completethe plan outlined in Section 7.1. We follow the argument of [F4]. We firstshow that this ceter is canonically isomorphic to the so-called classical W-algebra associated to the Langlands dual Lie algebra Lg. Next, we prove thatthe classical W-algebra is in turn isomorphic to the algebra of functions onthe space of LG-opers on the formal disc D. The algebra Fun OpLG(D) offunctions on the space of LG-opers on D carries a vertex Poisson structure.We show that our isomorphism between the center of Vκc(g) (with its canon-ical vertex Poisson structure coming from the deformation of the level) andFunOpLG(D) preserves vertex Poisson structures. Using this isomorphism,we show that the center of the completed enveloping algebra of g at the crit-ical level is isomorphic, as a Poisson algebra, to the algebra FunOpLG(D×)of functions on the space of opers on the punctured disc D×. The latterisomorphism was conjectured by V. Drinfeld.

Here is a more detailed description of the material of this chapter.In Section 8.1 we identify the center of the vertex algebra Vκc

(g) with theintersection of the kernels of certain operators, which we identify in turn withthe classical W-algebra associated to the Langlands dual Lie algebra Lg. Wealso show that the corresponding vertex Poisson algebra structures coincide.In Section 8.2.1 we define the Miura opers and explain their relationship tothe connections on a certain H-bundle. We then show in Section 8.2 thatthis classicalW-algebra (resp., the commutative vertex algebra π0) is nothingbut the algebra of functions on the space of LG-opers on the disc (resp.,Miura LG-opers on the disc). Furthermore, the embedding of the classicalW-algebra into π0 coincides with the Miura map between the two algebrasof functions. Finally, in Section 8.3 we consider the center of the completeduniversal enveloping algebra of g at the critical level. We identify it with thealgebra of functions on the space of LG-opers on the punctured disc. We thenprove that this identification satisfies various compatibilities. In particular, weshow that an affine analogue of the Harish-Chandra homomorphism obtained

214

8.1 Description of the center of the vertex algebra 215

by evaluating central elements on the Wakimoto modules is nothing but theMiura transformation from Miura LG-opers to LG-opers on the punctureddisc.

8.1 Description of the center of the vertex algebra

In this section we use Proposition 7.3.6 to describe the center z(g) of Vκc(g)as defined in Section 3.3.2.

According to Proposition 7.3.6, z(g) is identified with a subspace in theintersection of the kernels of the operators V i[1], i = 1, . . . , `, in π0. We nowshow that z(g) is actually equal to this intersection. This is Step 5 of the plandescribed in Section 7.1.

8.1.1 Computation of the character of z(g)

As explained in Section 3.3.3, the gκc -module Vκc(g) has a natural filtrationinduced by the Poincare–Birkhoff–Witt filtration on the universal envelopingalgebra U(gκc), and the associated graded space grVκc(g) is isomorphic to

Sym g((t))/g[[t]] ' Fun g∗[[t]],

where we use the following (coordinate-dependent) pairing

〈A⊗ f(t), B ⊗ g(t)〉 = 〈A,B〉Rest=0 f(t)g(t)dt (8.1.1)

for A ∈ g∗ and B ∈ g.Recall that z(g) is equal to the space of g[[t]]-invariants in Vκc(g). The sym-

bol of a g[[t]]-invariant vector in Vκc(g) is a g[[t]]-invariant vector in grVκc

(g),i.e., an element of the space of g[[t]]-invariants in Fun g∗[[t]]. Hence we obtainan injective map

gr z(g) → (Fun g∗[[t]])g[[t]] = Inv g∗[[t]] (8.1.2)

(see Lemma 3.3.1).According to Theorem 3.4.2, the algebra Inv g∗[[t]] is the free polynomial

algebra in the generators P i,n, i = 1, . . . , `;n < 0 (see Section 3.3.4 for thedefinition of the polynomials P i,n).

We have an action of the operator L0 = −t∂t on Fun g∗[[t]]. It defines aZ-gradation on Fun g∗[[t]] such that deg J

a

n = −n. Then degP i,n = di − n,and according to formula (4.3.3), the character of Inv g∗[[t]] is equal to

ch Inv g∗[[t]] =∏i=1

∏ni≥di+1

(1− qni)−1. (8.1.3)

We now show that the map (8.1.2) is an isomorphism using Proposition 6.3.3.Consider the Lie subalgebra b+ = (b+ ⊗ 1) ⊕ (g ⊗ tC[[t]]) of g[[t]]. The

216 Identification of the center with functions on opers

natural surjective homomorphism M0,κc → Vκc(g) gives rise to a map of thecorresponding subspaces of b+-invariants

φ : (M0,κc)eb+ → Vκc(g)eb+ .

Both M0,κc and Vκc(g) have natural filtrations (induced by the PBW filtra-tion on Uκc

(g)) which are preserved by the homomorphism between them.Therefore we have the corresponding map of associated graded

φcl : (gr M0,κc)eb+ → (grVκc(g))eb+ .

Since Vκc(g) is a direct sum of finite-dimensional representations of the con-stant subalgebra g ⊗ 1 of g[[t]], we find that any b+-invariant in Vκc

(g) ofgrVκc

(g) is automatically a g[[t]]-invariant. Therefore we obtain that

Vκc(g)eb+ = Vκc(g)g[[t]],

(grVκc(g))eb+ = (grVκc(g))g[[t]] = C[P i,m]i=1,...,`;m<0.

We need to describe (gr M0,κc)eb+ . First, observe that

gr M0,κc = Sym g((t))/b+ ' Fun g∗[[t]](−1),

where

g∗[[t]](−1) = ((n−)∗ ⊗ t−1)⊕ g∗[[t]] ' (g((t))/b+)∗,

and we use the pairing (8.1.1) between g((t)) and g∗((t)). In terms of thisidentification, the map φcl becomes a ring homomorphism

Fun g∗[[t]](−1) → Fun g∗[[t]]

induced by the natural embedding g∗[[t]]→ g∗[[t]](−1).Suppose that our basis Ja of g is chosen in such a way that it is the

union of two subsets, which constitute bases in b+ and in n−. Let Ja

n be thepolynomial function on n

(−1)+ defined by the formula

Ja

n(A(t)) = Rest=0〈A(t), Ja〉tndt. (8.1.4)

Then, as an algebra, Fun g∗[[t]](−1) is generated by these linear functionalsJ

a

n, where n < 0, or n = 0 and Ja ∈ n−.Next, we construct b+-invariant functions on g∗[[t]](−1) in the same way as

in Section 3.3.4, by substituting the generating functions

Ja(z) =

∑n

Ja

nz−n−1

(with the summation over n < 0 or n ≤ 0 depending on whether Ja ∈ b+ orn−) into the invariant polynomials P i, i = 1, . . . , ` on g. But since now J

a(z)

has non-zero z−1 coefficients if Ja ∈ n−, the resulting series

P i(Ja(z)) =

∑m∈Z

P i,mz−m−1


will have non-zero coefficients P i,m for all m < di. Thus, we obtain a naturalhomomorphism

C[Pi,mi ]i=1,...,`;mi<di → (Fun g∗[[t]](−1))eb+ .

Lemma 8.1.1 This homomorphism is an isomorphism.

Proof. Let

g∗[[t]](0) = ((n−)∗ ⊗ 1)⊕ (g∗ ⊗ tC[[t]]) = tg∗[[t]](−1).

Clearly, the spaces of b+-invariant functions on g∗[[t]](−1) and g∗[[t]](0) areisomorphic (albeit the Z-gradings are different), so we will consider the latterspace.

Denote by g∗[[t]]reg(0) the intersection of g∗[[t]](0) and

Jg∗reg = g∗reg × (g∗ ⊗ tC[[t]]).

Thus, g∗[[t]]reg(0) = ((n−)∗,reg⊗1)⊕ (g∗⊗ tC[[t]]), where (n−)∗,reg = (n−)∗∩g∗regis an open dense subset of (n−)∗, so that nreg

+ is open and dense in n+.Recall the morphism Jp : Jg∗reg → JP introduced in the proof of Theo-

rem 3.4.2. It was shown there that the group JG = G[[t]] acts transitivelyalong the fibers of Jp. Let B+ be the subgroup of G[[t]] corresponding to theLie algebra b+ ⊂ g[[t]]. Note that for any x ∈ g∗[[t]]reg(0) , the group B+ is equalto the subgroup of all elements g of G[[t]] such that g ·x ∈ g∗[[t]]reg(0) . Therefore

B+ acts transitively along the fibers of the restriction of the morphism Jp tog∗[[t]]reg(0) .

This implies, in the same way as in the proof of Theorem 3.4.2, that thering of B+-invariant (equivalently, b+-invariant) polynomials on g∗[[t]]reg(0) isthe ring of functions on the image of g∗[[t]]reg(0) in JP under the map Jp. But itfollows from the construction that the image of g∗[[t]]reg(0) in JP is the subspacedetermined by the equations P i,m = 0, i = 1, . . . , `;m = −1. Hence the ring ofb+-invariant polynomials on g∗[[t]]reg(0) is equal to C[P i,mi

]i=1,...,`;mi<−1. Sinceg∗[[t]]reg(0) is dense in g∗[[t]](0), we obtain that this is also the ring of invariantpolynomials on n+.

When we pass from g∗[[t]](0) to g∗[[t]](−1), we need to take into accountthe shifting of the indices P i,mi 7→ P i,mi+di+1 of invariant polynomials cor-responding to the shift J

a

n 7→ Ja

n+1. Then we obtain the statement of thelemma.

Corollary 8.1.2 The map φcl is surjective.

Proof. The map φcl corresponds to taking the quotient of the free poly-nomial algebra on Pi,mi

, i = 1, . . . , `;mi < di, by the ideal generated byPi,mi

, i = 1, . . . , `; 0 ≤ mi < di.


It follows from the construction that degP i,m = m − di. Hence we obtainthe following formula for the character of (gr M0,κc)

eb+ = (Fun n(−1)+ )eb+ :

ch (gr M0,κc)eb+ =

∏m>0

(1− qm)−`.

Now recall that by Theorem 6.3.3 the Verma module M0,κc is isomorphicto the Wakimoto module W+

0,κc. Hence (M0,κc)

eb+ = (W+0,κc

)eb+ . In addition,

according to Lemma 6.3.4 we have (W+0,κc

)eb+ = π0, and so its character is

also equal to∏m>0

(1− qm)−`. Therefore we find that the natural embedding

gr(Meb+0,κc

) → (gr M0,κc)eb+

is an isomorphism.Consider the commutative diagram

gr(Meb+0,κc

) −−−−→ gr(Vκc(g)g[[t]])y y

(gr M0,κc)eb+ −−−−→ (grVκc(g))g[[t]].

It follows from the above discussion that the left vertical arrow is an isomor-phism. Moreover, by Corollary 8.1.2 the lower horizontal arrow is surjective.Therefore the right vertical arrow is surjective. But it is also injective, ac-cording to Lemma 3.3.1. Therefore we obtain an isomorphism

gr(Vκc(g)g[[t]]) ' (grVκc(g))g[[t]].

In particular, this implies that the character of gr z(g) = gr(Vκc(g))g[[t]] isequal to that of (grVκc

(g)g[[t]]) given by formula (4.3.3). Since ch z(g) =ch(gr z(g)), we find that

ch z(g) =∏i=1

∏ni≥di+1

(1− qni)−1. (8.1.5)

Thus, we obtain the following (see Section 4.3.1):

Theorem 8.1.3 The center z(g) is “as large as possible,” i.e.,

gr z(g) = Inv g∗[[t]].

Thus, there exist central elements Si ∈ z(g) ⊂ Vκc(g) whose symbols are equalto P i,−1 ∈ Inv g∗[[t]], i = 1, . . . , `, and such that

z(g) = C[Si,(n)]i=1,...,`;n<0|0〉,

where the Si,(n)’s are the Fourier coefficients of the vertex operator Y (Si, z).


This is a non-trivial result which tells us a lot about the structure of the cen-ter. But we are not satisfied with it, because, as explained in Section 3.5.2, wewould like to understand the geometric meaning of the center and in particularwe want to know how the group AutO acts on z(g). This is expressed in The-orem 4.3.2, which identifies z(g) with the algebra of functions on OpLG(D)(note that Theorem 4.3.2 implies Theorem 8.1.3, see formula (4.3.2)). Toprove this, we need to work a little harder and complete Steps 5 and 6 of ourplan presented in Section 7.1.

8.1.2 The center and the classical W-algebra

According to Proposition 7.3.6, z(g) is contained in the intersection of thekernels of the operators V i[1], i = 1, . . . , `, on π0. Now we compute thecharacter of this intersection and compare it with the character formula (8.1.5)for z(g) to show that z(g) is equal to the intersection of the kernels of theoperators V i[1].

First, we identify this intersection with a classical limit of a one-parameterfamily of vertex algebras, called the W-algebras. The W-algebra Wν(g) as-sociated to a simple Lie algebra g and an invariant inner product ν on g wasdefined in [FF6] (see also [FB], Chapter 15) via the quantum Drinfeld–Sokolovreduction. For generic values of ν the vertex algebraWν(g) is equal to the in-tersection of the kernels of certain screening operators in a Heisenberg vertexalgebra. Let us recall the definition of these operators.†

Consider another copy of the Heisenberg Lie algebra hν introduced in Sec-tion 6.2.1. To avoid confusion, we will denote the generators of this HeisenbergLie algebra by

bi,n, i = 1, . . . , `;n ∈ Z.

We have the vertex operator

V ν−αi

(z) : πν0 → πν

−αi

defined by formula (7.2.13), and let V ν−αi

[1] =∫V ν−αi

(z)dz be its residue. Wecall it a W-algebra screening operator (to distinguish it from the Kac–Moody algebra screening operators defined above). Since

V ν−αi

(z) = Yπν0 ,πν

−αi(| − αi〉, z)

(in the notation of Section 7.2.1), we obtain that the intersection of the kernelsof V ν

−αi[1], i = 1, . . . , `, in πν

0 is a vertex subalgebra of πν0 . By Theorem 15.4.12

of [FB], for generic values of ν theW-algebraWν(g) is equal to the intersectionof the kernels of the operators V ν

−αi[1], i = 1, . . . , `, in πν

0 .We are interested in the limit of Wν(g) when ν →∞. To define this limit,

we fix an invariant inner product ν0 on g and denote by ε the ratio between

† In fact, we may use this property to define Wν(g) for all ν, see [FF7].


ν0 and ν. We have the following formula for the ith simple root αi ∈ h∗ as anelement of h using the identification between h∗ and h induced by ν = ν0/ε:

αi = ε2

ν0(hi, hi)hi.

Let

b′i,n = ε2

ν0(hi, hi)bi,n, (8.1.6)

where the bi,n’s are the generators of hν . Consider the C[ε]-lattice in πν0⊗CC[ε]

spanned by all monomials in b′i,n, i = 1, . . . , `;n < 0. We denote by π∨0 thespecialization of this lattice at ε = 0; it is a commutative vertex algebra.

In the limit ε → 0, we obtain the following expansion of the operatorV ν−αi

[1]:

V ν−αi

[1] = ε2

ν0(hi, hi)Vi[1] + . . . ,

where the dots denote terms of higher order in ε, and the operator Vi[1] actingon π∨0 is given by the formula

Vi[1] =∑m≤0

Vi[m] Db′i,m−1, (8.1.7)

where

Db′i,m· b′j,n = aijδn,m, (8.1.8)

(aij) is the Cartan matrix of g, and

∑n≤0

Vi[n]z−n = exp

(−∑m>0

b′i,−m

mzm

).

The intersection of the kernels of the operators Vi[1], i = 1, . . . , `, is acommutative vertex subalgebra of π∨0 , which we denote by W(g) and call theclassical W-algebra associated to g.

Note that the structure of commutative vertex algebra on π∨0 is independentof the choice of ν0. The operators Vi[1] get rescaled if we change ν0 and sotheir kernels are independent of ν0. Therefore W(g) is a commutative vertexsubalgebra of π∨0 that is independent of ν0.†

8.1.3 The appearance of the Langlands dual Lie algebra

Comparing formulas (7.3.13) and (8.1.7), we find that after the substitutionbi,n 7→ −b′i,n, the operators V i[1] become the operators Vi[1], except thatthe matrix coefficient aji in formula (7.3.14) gets replaced by aij in formula

† However, as we will see below, both π∨0 and W(g) also carry vertex Poisson algebrastructures, and those structures do depend on ν0.


(8.1.8). This is not a typo! There is a serious reason for that: while theoperator V i[1] was obtained as the limit of a vertex operator correspondingto the ith coroot of g, the operator Vi[1] was obtained as the limit of a vertexoperator corresponding to minus the ith root of g.

Under the exchange of roots and coroots, the Cartan matrix gets trans-posed. The transposed Cartan matrix of a simple Lie algebra g is the Cartanmatrix of another simple Lie algebra; namely, the Langlands dual Lie al-gebra of g, which is denoted by Lg. Because of this transposition, we mayidentify canonically the Cartan subalgebra Lh of Lg with the dual space h∗ tothe Cartan subalgebra h of g, so that the simple roots of g (which are vectorsin h∗) become the simple coroots of Lg (which are vectors in Lh).

Let us identify π0 corresponding to g with π∨0 corresponding to Lg by send-ing bi,n 7→ −b′i,n. Then the operator V i[1] attached to g becomes the operatorVi[1] attached to Lg. Therefore the intersection of the kernels of the operatorsV i[1], i = 1, . . . , `, on π0, attached to a simple Lie algebra g, is isomorphicto the intersection of the kernels of the operators Vi[1], i = 1, . . . , `, on π∨0 ,attached to Lg.

Using Proposition 7.3.6, we now find that z(g) is embedded into the inter-section of the kernels of the operators Vi[1], i = 1, . . . , `, on π∨0 , i.e., into theclassicalW-algebra W(Lg). Furthermore, we have the following result, whoseproof will be postponed till Section 8.2.4 below.

Lemma 8.1.4 The character of W(Lg) is equal to the character of z(g) givenby formula (8.1.5).

Therefore we obtain the following result.

Theorem 8.1.5 The center z(g) is isomorphic, as a commutative vertex al-gebra, to the intersection of the kernels of the operators V i[1], i = 1, . . . , `, onπ0, and hence to the classical W-algebra W(Lg).

This completes Step 5 of the plan presented in Section 7.1.

8.1.4 The vertex Poisson algebra structures

In addition to the structures of commutative vertex algebras, both z(g) andW(Lg) also carry the structures of vertex Poisson algebra, and we wishto show that the isomorphism of Theorem 8.1.5 is compatible with thesestructures.

We will not give a precise definition of vertex Poisson algebras here, referringthe reader to [FB], Section 16.2. We recall that a vertex Poisson algebra Pis in particular a vertex Lie algebra, and so we attach to it an ordinary Liealgebra

Lie(P ) = P ⊗ C((t))/ Im(T ⊗ 1 + 1⊗ ∂t)


(see [FB], Section 16.1.7).According to Proposition 16.2.7 of [FB], if Vε is a one-parameter family

of vertex algebras, then the center Z(V0) of V0 acquires a natural vertexPoisson algebra structure. Namely, at ε = 0 the polar part of the operationY , restricted to Z(V0), vanishes, so we define the operation Y− on Z(V0) asthe ε-linear term of the polar part of Y .

Let us fix a non-zero inner product κ0 on g, and let ε be the ratio betweenthe inner products κ−κc and κ0. Consider the vertex algebras Vκ(g) as a one-parameter family using ε as a parameter. Then we obtain a vertex Poissonstructure on z(g), the center of Vκc(g) (corresponding to ε = 0). We willdenote z(g), equipped with this vertex Poisson structure, by z(g)κ0 .

Next, consider the Heisenberg vertex algebra πκ−κc0 introduced in Sec-

tion 6.2.1. From now on, to avoid confusion, we will write πκ−κc0 (g) to in-

dicate that it is associated to the Cartan subalgebra h of g. Let ε be definedby the formula κ − κc = εκ0 as above. We define the commutative vertexalgebra π0(g) as an ε→ 0 limit of πκ−κc

0 (g) in the following way. Recall thatπκ−κc

0 (g) is the Fock representation of the Heisenberg Lie algebra with gen-erators bi,n, i = 1, . . . , `;n ∈ Z. The commutation relations between them areas follows:

[bi,n, bj,m] = εnκ0(hi, hj)δn,−m. (8.1.9)

Consider the C[ε]-lattice in πκ−κc0 ⊗CC[ε] spanned by all monomials in bi,n, i =

1, . . . , `;n < 0. Now π0(g) is by definition the specialization of this lattice atε = 0.

This is a commutative vertex algebra, but since it is defined as the limit of aone-parameter family of vertex algebras, it acquires a vertex Poisson structure.This vertex Poisson structure is uniquely determined by the Poisson brackets

bi,n, bj,m = nκ0(hi, hj)δn,−m

in Lie(π0(g)), which immediately follow from formula (8.1.9). In particular,this vertex Poisson structure depends on κ0, and so we will write π0(g)κ0 toindicate this dependence.

Recall that the homomorphism wκc : Vκc(g)→W0,κc = Mg⊗π0(g) may bedeformed to a homomorphism wκ : Vκ(g)→W0,κ = Mg⊗πκ−κc

0 (g). Thereforethe ε-linear term of the polar part of the operation Y of Vκ(g), restricted toz(g), which is used in the definition of the vertex Poisson structure on z(g),may be computed by restricting to z(g) ⊂ π0(g) the ε-linear term of the polarpart of the operation Y of πκ−κc

0 (g) ⊂ W0,κ. But the latter gives π0(g) thestructure of a vertex Poisson algebra, which we denote by π0(g)κ0 . Thereforewe obtain the following:

Lemma 8.1.6 The embedding z(g)κ0 → π0(g)κ0 is a homomorphism of vertex


Poisson algebras. The corresponding map of local Lie algebras Lie(z(g)κ0) →Lie(π0(g)κ0) is a Lie algebra homomorphism.

On the other hand, consider the commutative vertex algebra π∨0 (g) andits subalgebra W(g) defined in Section 8.1.2. The vertex algebra π∨0 (g) wasdefined as the limit of a one-parameter family of vertex algebras, namely,πν

0 (g), where ν = ν0/ε, as ε → 0. Therefore πν0 (g) also carries a vertex

Poisson algebra structure. We will denote the resulting vertex Poisson algebraby π∨0 (g)ν0 . Its subalgebra W(g) is obtained as the limit of a one-parameterfamily of vertex subalgebras of πν

0 (g); namely, of Wν(g). Therefore W(g) isa vertex Poisson subalgebra of π∨0 (g)ν0 , which we will denote by W(g)ν0 .

Thus, we see that the objects appearing in the isomorphism of Theo-rem 8.1.5 carry natural vertex Poisson algebra structures. We claim thatthis isomorphism is in fact compatible with these structures.

Let us explain this more precisely. Note that the restriction of a non-zeroinvariant inner product κ0 on g to h defines a non-zero inner product on h∗,which is the restriction of an invariant inner product κ∨0 on Lg. To avoidconfusion, let us denote by π0(g)κ0 and π∨0 (Lg)κ∨0

the classical limits of theHeisenberg vertex algebras defined above, with their vertex Poisson structurescorresponding to κ0 and κ∨0 , respectively. We have an isomorphism of vertexPoisson algebras

ı : π0(g)κ0

∼→ π∨0 (Lg)κ∨0,

bi,n 7→ −b′i,n,

where b′i,n is given by formula (8.1.6). The restriction of the isomorphism ı tothe subspace

⋂1≤i≤` KerV i[1] of π0(g)κ0 gives us an isomorphism of vertex

Poisson algebras ⋂1≤i≤`

KerV i[1] '⋂

1≤i≤`

KerV∨i [1],

where by V∨i [1] we denote the operator (8.1.7) attached to Lg. Recall thatwe have the following isomorphisms of vertex Poisson algebras:

z(g)κ0 '⋂

1≤i≤`

KerV i[1],

W(Lg)κ∨0'

⋂1≤i≤`

KerV∨i [1].

Therefore we obtain the following stronger version of Theorem 8.1.5:


Theorem 8.1.7 There is a commutative diagram of vertex Poisson algebras

π0(g)κ0

∼−−−−→ π∨0 (Lg)κ∨0x xz(g)κ0

∼−−−−→ W(Lg)κ∨0

(8.1.10)

8.1.5 AutO-module structures

Both z(g)κ0 and W(Lg)κ∨0carry actions of the group AutO, and we claim that

the isomorphism of Theorem 8.1.7 intertwines these actions. To see tist, wedescribe the two actions as coming from the vertex Poisson algebra structures.

In both cases the action of the group AutO is obtained by exponentiationof the action of the Lie algebra Der0O ⊂ DerO. In the case of the centerz(g)κ0 , the action of DerO is the restriction of the natural action on Vκc

(g)which comes from its action on gκc

(preserving its Lie subalgebra g[[t]]) byinfinitesimal changes of variables. But away from the critical level, i.e., whenκ 6= κc, the action of DerO is obtained through the action of the Virasoroalgebra which comes from the conformal vector Sκ, given by formula (6.2.5),which we rewrite as follows:

Sκ =κ0

κ− κcS1,

where S1 is given by formula (3.1.1) and κ0 is the inner product used in thatformula. Thus, the Fourier coefficients Ln, n ≥ −1, of the vertex operator

Y (Sκ, z) =∑n∈Z

Lnz−n−2

generate the DerO-action on Vκ(g) when κ 6= κc.

In the limit κ → κc, we have Sκ = ε−1S1, where as before ε =κ− κc

κ0.

Therefore the action of DerO is obtained through the vertex Poisson operationY− on z(g)κ0 , defined as the limit of ε−1 times the polar part of Y when ε→ 0(see [FB], Section 16.2), applied to S1 ∈ z(g)κ0 .

In other words, the DerO-action is generated by the Fourier coefficientsLn, n ≥ −1, of the series

Y−(S1, z) =∑

n≥−1

Lnz−n−2.

Thus, we see that the natural DerO-action (and hence AutO-action) on z(g)κ0

is encoded in the vector S1 ∈ z(g)κ0 through the vertex Poisson algebra struc-ture on z(g)κ0 . Note that this action endows z(g)κ0 with a quasi-conformalstructure (see Section 6.2.4).

Likewise, there is a DerO-action on W(Lg)κ∨0coming from its vertex Pois-

son algebra structure. The vector generating this action is also equal to the


limit of a conformal vector in the W-algebra Wν(Lg) (which is a conformalvertex algebra) as ν →∞. This conformal vector is unique because as we seefrom the character formula for Wν(Lg) given in the right hand side of (8.1.5)the homogeneous component of Wν(Lg) of degree two (where all conformalvectors live) is one-dimensional.† The limit of this vector as ν → ∞ givesrise to a vector in W(Lg)κ∨0

, which we denote by t, such that the Fouriercoefficients of Y−(t, z) generate a DerO-action on W(Lg)κ∨0

(in the next sec-tion we will explain the geometric meaning of this action). Since such avector is unique, it must be the image of S1 ∈ z(g)κ0 under the isomorphismz(g)κ0 'W(Lg)κ∨0

.Therefore we conclude that the DerO-actions (and hence the corresponding

AutO-actions) on both z(g)κ0 and W(Lg)κ∨0are encoded, via the respective

vertex Poisson structures, by certain vectors, which are in fact equal to theclassical limits of conformal vectors. Under the isomorphism of Theorem 8.1.7these vectors are mapped to each other. Thus, we obtain the following:

Proposition 8.1.8 The commutative diagram (8.1.10) is compatible with theactions of DerO and AutO.

We have already calculated in formula (6.2.5) the image of Sκ in W0,κ whenκ 6= κc. By passing to the limit κ → κc we find that the image of S1 = εSκ

belongs to π0(g)κ0 ⊂W0,κ and is equal to

12

∑i=1

bi,−1bi−1 − ρ−2,

where bi and bi are dual bases with respect to the inner product κ0 used inthe definition of S1, restricted to h, and ρ is the element of h corresponding toρ ∈ h∗ under the isomorphism h∗ ' h induced by κ0. Under the isomorphismof Theorem 8.1.7, this vector becomes the vector S1 ∈ z(g)κ0 ⊂ π0(g)κ0 , whichis responsible for the DerO action on it.

The action of the corresponding operators Ln ∈ DerO, n ≥ −1, on π0(g)κ0

is given by derivations of the algebra structure which are uniquely determinedby formulas (6.2.13). Therefore the action of the Ln’s on π∨0 (Lg)κ∨0

is as follows(recall that bi,n 7→ −b′i,n under our isomorphism π0(g)κ0 ' π∨0 (Lg)κ∨0

):

Ln · b′i,m = −mb′i,n+m, −1 ≤ n < −m,Ln · b′i,−n = −n(n+ 1), n > 0, (8.1.11)

Ln · b′i,m = 0, n > −m.

These formulas determine the DerO-action on π∨0 (Lg)κ∨0. By construction,

W(Lg)κ∨0⊂ π∨0 (Lg)κ∨0

is preserved by this action.

† This determines this vector up to a scalar, which is fixed by the commutation relations.


Note in particular that the above actions of DerO on π0(g)κ0 and π∨0 (Lg)κ∨0

(and hence on z(g)κ0 and W(Lg)κ∨0) are independent of the inner product κ0.

8.2 Identification with the algebra of functions on opers

We have now identified the center z(g)κ0 with the classicalW-algebra W(Lg)κ∨0

in a way compatible with the vertex Poisson algebra structures and the(DerO,AutO)-actions. The last remaining step in our proof of Theorem 4.3.2(Step 6 of our plan from Section 7.1) is the identification of W(Lg)κ∨0

withthe algebra OpLG(D) of functions on the space of LG-opers on the disc. Thisis done in this section.

We start by introducing Miura opers and a natural map from generic Miuraopers to opers called the Miura transformation. We then show that the al-gebras of functions on G-opers and generic Miura G-opers on the disc D areisomorphic to W(g)ν0 and π∨0 (g)ν0 , respectively. Furthermore, the homo-morphism W(g)ν0 → π∨0 (g)ν0 corresponding to the Miura transformation isprecisely the embedding constructed in Section 8.1.2. This will enable us toidentify the center z(g) with the algebra of functions on LG-opers on D.

8.2.1 Miura opers

Let G be a simple Lie group of adjoint type.A Miura G-oper on X, which is a smooth curve, or D, or D×, is by

definition a quadruple (F ,∇,FB ,F ′B), where (F ,∇,FB) is a G-oper on X

and F ′B is another B-reduction of F which is preserved by ∇.

Consider the space MOpG(D) of Miura G-opers on the disc D. A B-reduction of F which is preserved by the connection ∇ is uniquely determinedby a B-reduction of the fiber F0 of F at the origin 0 ∈ D (recall that theunderlying G-bundles of all G-opers are isomorphic to each other). The set ofsuch reductions is the F0-twist (G/B)F0 of the flag manifold G/B. Let Funiv

be the universal G-bundle on OpG(D) whose fiber at the oper (F ,∇,FB) isF0. Then we obtain that

MOpG(D) ' (G/B)Funiv = Funiv ×GG/B.

A Miura G-oper is called generic if the B-reductions FB and F ′B are ingeneric relative position. We denote the space of generic Miura opers on D

by MOpG(D)gen. We have a natural forgetful morphism MOpG(D)gen →OpG(D). The group

NFB,0 = FB,0 ×BN,

where FB,0 is the fiber of FB at 0, acts on (G/B)F0 , and the subset of generic

8.2 Identification with the algebra of functions on opers 227

reductions is the open NFB,0-orbit of (G/B)F0 . This orbit is in fact an NFB,0-torsor. Therefore we obtain that the space of generic Miura opers with a fixedunderlying G-oper (F ,∇,FB) is a principal NFB,0-bundle.

This may be rephrased as follows. Let FB,univ be the B-reduction of theuniversal G-bundle Funiv, whose fiber at the oper (F ,∇,FB) is FB,0. Then

MOpG(D)gen = FB,univ ×BU , (8.2.1)

where U ' N is the open B-orbit in G/B.Now we identify MOpG(D)gen with the space of H-connections. Consider

the H-bundles FH = FB/N and F ′H = F ′B/N corresponding to a genericMiura oper (F ,∇,FB , F ′B) on X. If P is an H-bundle, then applying to itthe automorphism w0 of H, corresponding to the longest element of the Weylgroup of G, we obtain a new H-bundle, which we denote by w∗0(P).

Lemma 8.2.1 For a generic Miura oper (F ,∇,FB ,F ′B) the H-bundle F ′H isisomorphic to w∗0(FH).

Proof. Consider the vector bundles gF = F ×G

g, bFB= FB×

Bb and bF ′B =

F ′B ×B

b. We have the inclusions bFB, bF ′B ⊂ gF which are in generic position.

Therefore the intersection bFB∩bF ′B is isomorphic to bFB

/[bFB, bFB

], which isthe trivial vector bundle with fiber h. It naturally acts on the bundle gF andunder this action gF decomposes into a direct sum of h and the line subbun-dles gF,α, α ∈ ∆. Furthermore, bFB

=⊕

α∈∆+gF,α, bF ′B =

⊕α∈∆+

gF,w0(α).Since the action of B on n/[n, n] factors through H = B/N , we find that

FH ×H

⊕i=1

Cαi'⊕i=1

gF,αi, F ′H ×

H

⊕i=1

Cαi '⊕i=1

gF,w0(αi).


FH ×H

Cαi ' F ′H ×H

Cw0(αi), i = 1, . . . , `.

Since G is of adjoint type by our assumption, the above associated line bundlescompletely determine FH and F ′H , and the above isomorphisms imply thatF ′H ' w∗0(FH).

Since the B-bundle F ′B is preserved by the oper connection ∇, we obtaina connection ∇ on F ′H and hence on FH . But according to Lemma 4.2.1,we have FH ' Ωρ. Therefore we obtain a morphism β from the spaceMOpG(D)gen of generic Miura opers on D to the space Conn(Ωρ)D of con-nections on the H-bundle Ωρ on D.

Explicitly, connections on the H-bundle Ωρ may be described by the oper-ators

∇ = ∂t + u(t), u(t) ∈ h[[t]],


where t is a coordinate on the disc D, which we use to trivialize Ωρ. Let s bea new coordinate such that t = ϕ(s). Then the same connection will appearas ∂s + u(s), where

u(s) = ϕ′ · u(ϕ(s))− ρϕ′′

ϕ′. (8.2.2)

This formula describes the action of the group AutO (and the Lie algebraDerO) on Conn(Ωρ)D.

Proposition 8.2.2 The map β : MOpG(D)gen → Conn(Ωρ)D is an isomor-phism.

Proof. We define a map τ in the opposite direction. Suppose we are givena connection ∇ on the H-bundle Ωρ on D. We associate to it a generic Miuraoper as follows. Let us choose a splitting H → B of the homomorphismB → H and set F = Ωρ ×

HG,FB = Ωρ ×

HB, where we consider the adjoint

action of H on G and on B obtained through the above splitting. The choiceof the splitting also gives us the opposite Borel subgroup B−, which is theunique Borel subgroup in generic position with B containing H. Then we setF ′B = Ωρ ×

HB−w0B.

Observe that the space of connections on F is isomorphic to the directproduct

Conn(Ωρ)D ×⊕α∈∆

ωα(ρ)+1.

Its subspace corresponding to negative simple roots is isomorphic to the space(⊕ì=1 g−αi

)⊗ O. Having chosen a basis element fi of g−αi for each i =

1, . . . , `, we now construct an element p−1 =∑`

i=1 fi of this space. Now weset ∇ = ∇+p−1. By construction, ∇ has the correct relative position with theB-reduction FB and preserves the B-reduction F ′B . Therefore the quadruple(F ,∇,FB ,F ′B) is a generic Miura oper on D. We set τ(∇) = (F ,∇,FB ,F ′B).

This map is independent of the choice of splitting H → B and of thegenerators fi, i = 1, . . . , `. Indeed, changing the splitting H → B amountsto a conjugation of the old splitting by an element of N . This is equivalentto applying to ∇ the gauge transformation by this element. Therefore itwill not change the underlying Miura oper structure. Likewise, rescaling ofthe generators fi may be achieved by a gauge transformation by a constantelement of H, and this again does not change the Miura oper structure. It isclear from the construction that β and τ are mutually inverse isomorphisms.

Under the isomorphism of Proposition 8.2.2, the natural forgetful morphism

MOpG(D)gen → OpG(D)


becomes a map

µ : Conn(Ωρ)D → OpG(D). (8.2.3)

We call this map the Miura transformation.The Miura transformation (8.2.3) gives rise to a homomorphism of the

corresponding rings of functions µ : FunOpG(D) → FunConn(Ωρ)D. Eachspace has an action of DerO and µ is a DerO-equivariant homomorphismbetween these rings. We will now identify Fun Conn(Ωρ)D with π∨0 (g) andthe image of µ with the intersection of kernels of the W-algebra screeningoperators. This will give us an identification of Fun OpG(D) with W(g).

8.2.2 Explicit realization of the Miura transformation

As explained in Section 4.2.4, if we choose a coordinate t on the disc, we canrepresent each oper connection in the canonical form

∂t + p−1 +∑i=1

vi(t) · ci, vi(t) ∈ C[[t]]

(see Section 4.2.4), where

vi(t) =∑n<0

vi,nt−n−1.

Thus,

FunOpG(D) = C[vi,ni]i=1,...,`;ni<0.

If we choose a Cartan subalgebra h in b, then, according to Proposition 8.2.2,we can represent each generic Miura oper by a connection operator of the fol-lowing type:

∂t + p−1 + u(t), u(t) ∈ h[[t]]. (8.2.4)

Set ui(t) = αi(u(t)), i = 1, . . . , `, and

ui(t) =∑n<0

ui,nt−n−1.

Then

FunMOpG(D)gen = FunConn(Ωρ)D = C[ui,n]i=1,...,`;n<0.

Hence the Miura transformation gives rise to a homomorphism

µ : C[vi,ni]i=1,...,`;ni<0 → C[ui,n]i=1,...,`;n<0. (8.2.5)

Example. We compute the Miura transformation µ in the case when g = sl2.In this case an oper has the form

∂t +(

0 v(t)1 0

),


and a generic Miura oper has the form

∂t +(

12u(t) 0

1 − 12u(t)

).

To compute µ, we need to find an element ofN [[t]] such that the correspondinggauge transformation brings the Miura oper into the oper form. We find that(

1 − 12u(t)

0 1

)(∂t +

(12u(t) 0

1 − 12u(t)

))(1 1

2u(t)0 1

)=

∂t +(

0 14u(t)

2 + 12∂tu(t)

1 0

).


µ(u(t)) = v(t) =14u(t)2 +

12∂tu(t),

which may also be written in the form

∂2t − v(t) =

(∂t +

12u(t)

)(∂t −

12u(t)

).

It is this transformation that was originally introduced by R. Miura as themap intertwining the flows of the KdV hierarchy and the mKdV hierarchy.

In the case when g = sl2 opers are projective connections and Miura opersare affine connections (see, e.g., [FB], Chapter 9). The Miura transformation isnothing but the natural map from affine connections to projective connections.

By construction, the Miura transformation (8.2.3) and hence the homomor-phism (8.2.5) are DerO-equivariant. The action of DerO on FunOpG(D) isobtained from formulas (4.2.5), and the action on the algebra FunConn(Ωρ)D

is obtained from formula (8.2.2). It translates into the following explicit for-mulas for the action of the generators Ln = −tn+1∂t, n ≥ −1, on the ui,m’s:

Ln · ui,m = −mui,n+m, −1 ≤ n < −m,Ln · ui,−n = −n(n+ 1), n > 0, (8.2.6)

Ln · ui,m = 0, n > −m.

8.2.3 Screening operators

Recall the realization (8.2.1) of MOpG(D)gen as an FB,univ-twist of the B-torsor U ⊂ G/B. As explained in Section 4.2.4, the B-torsors FB,0 may beidentified for all opers (F ,FB ,∇). Therefore we obtain that the group NFB,0

acts transitively on the fibers of the map MOpG(D)gen → OpG(D). Accordingto Proposition 8.2.2, we have an isomorphism MOpG(D)gen ' Conn(Ωρ)D.Therefore NFB,0 acts transitively along the fibers of the Miura transformationµ. Therefore we obtain that the image of the homomorphism µ is equal to


the space of NFB,0-invariants of FunConn(Ωρ)D, and hence to the space ofnFB,0-invariants of FunConn(Ωρ)D.

Let us fix a Cartan subalgebra h in b and a trivialization of FB,0. Using thistrivialization, we identify the twist nFB,0 with n. Now we choose the generatorsei, i = 1, . . . , `, of n with respect to the action of h on n in such a way thattogether with the previously chosen fi they satisfy the standard relations ofg. The NFB,0-action on Conn(Ωρ)D then gives rise to an infinitesimal actionof ei on Conn(Ωρ)D. We will now compute the corresponding derivation onFunConn(Ωρ)D.

The action of ei is given by the infinitesimal gauge transformation

δu(t) = [xi(t) · ei, ∂t + p−1 + u(t)], (8.2.7)

where xi(t) ∈ C[[t]] is such that xi(0) = 1, and the right hand side of formula(8.2.7) belongs to h[[t]]. It turns out that these conditions determine xi(t)uniquely. Indeed, the right hand side of (8.2.7) reads

xi(t) · αi − ui(t)xi(t) · ei − ∂txi(t) · ei.

Therefore it belongs to h[[t]] if and only if

∂txi(t) = −ui(t)xi(t). (8.2.8)

If we write

xi(t) =∑n≤0

xi,nt−n,

and substitute it into formula (8.2.8), we obtain that the coefficients xi,n

satisfy the following recurrence relation:

nxi,n =∑

k+m=n;k<0;m≤0

ui,kxi,m, n < 0.

We find from this formula that∑n≤0

xi,nt−n = exp

(−∑m>0

ui,−m

mtm

). (8.2.9)

Now we obtain that

δuj(t) = αj(δu(t)) = aijxi(t),

where (aij) is the Cartan matrix of g. In other words, the operator ei acts onthe algebra FunConn(Ωρ)D = C[ui,n] by the derivation

∑j=1

aij

∑n≥0

xi,n∂

∂uj,−n−1, (8.2.10)

where xi,n are given by formula (8.2.9).


Now, the image of FunOpG(D) under the Miura map µ is the algebra of n-invariant functions on Conn(Ωρ)D. These are precisely the functions that areannihilated by the generators ei, i = 1, . . . , `, of n, which are given by formula(8.2.10). Therefore we obtain the following characterization of Fun OpG(D)as a subalgebra of Fun Conn(Ωρ)D.

Proposition 8.2.3 The image of FunOpG(D) in FunConn(Ωρ)D under theMiura map µ is equal to the intersection of the kernels of the operators givenby formula (8.2.10) for i = 1, . . . , `.

8.2.4 Back to the W-algebras

Comparing formula (8.2.10) with formula (8.1.7) we find that if we replace b′i,nby ui,n in formula (8.1.7) for the W-algebra screening operator Vi[1], thenwe obtain formula (8.2.10). Therefore the intersection of the kernels of theoperators (8.2.10) is equal to the intersection of the kernels of the operatorsVi[1], i = 1, . . . , `. But the latter is the classical W-algebra W(g)ν0 . Hencewe obtain the following commutative diagram:

π∨0 (g)ν0

∼−−−−→ FunConn(Ωρ)Dx xW(g)ν0

∼−−−−→ FunOpG(D)

(8.2.11)

where the top arrow is an isomorphism of algebras given on generators by theassignment b′i,n 7→ ui,n.

We can also compute the character of W(g) ' FunOpG(D) and proveLemma 8.1.4. It follows from Proposition 8.2.2 that the action of the groupN on Conn(Ωρ)D is free. Therefore we obtain that

FunConn(Ωρ)D ' FunOpG(D)⊗ FunN,

as vector spaces. The action of the generators ei of the Lie algebra n is givenby the operators Vi[1], each having degree −1 with respect to the Z+-gradingintroduced above. Therefore a root generator eα ∈ n, α ∈ ∆+, acts as anoperator of degree −〈α, ρ〉. This means that the action of n preserves theZ-grading on FunConn(Ωρ)D if we equip n with the negative of the princi-pal gradation, for which deg eα = 〈α, ρ〉. With respect to this grading thecharacter of FunN is equal to∏

α∈∆+

(1− q〈α,ρ〉)−1 =∏i=1

di∏ni=1

(1− qni)−1.

On the other hand, the character of Fun Conn(Ωρ)D is equal to∏n>0

(1− qn)−`.


Therefore the character of FunOpG(D) is equal to

∏n>0

(1− qn)−`∏i=1

di∏ni=1

(1− qni)−1.

Thus, we obtain that the character of FunOpG(D) 'W(g)ν0 is given by theright hand side of formula (8.1.5). This proves Lemma 8.1.4.

We also obtain from the above isomorphism a vertex Poisson algebra struc-ture on FunOpG(D) (the latter structure may alternatively be defined bymeans of the Drinfeld–Sokolov reduction, as we show below). As before, thesestructures depend on the choice of inner product ν0, which we will sometimesuse as a subscript to indicate which Poisson structure we consider.

The above diagram gives us a geometric interpretation of the W-algebrascreening operators Vi[1]: they correspond to the action of the generators ei

of n on FunConn(Ωρ)D.Comparing formulas (8.2.6) and (8.1.11), we find that the top isomorphism

in (8.2.11) is DerO-equivariant. Since the subspaces W(g)ν0 and FunOpG(D)are stable under the DerO-action, we find that the bottom isomorphism isalso DerO-equivariant.†

Furthermore, the action of DerO on all of the above algebras is independentof the inner product ν0. Thus we obtain the following:

Theorem 8.2.4 The diagram (8.2.11) is compatible with the action of DerO.

8.2.5 Completion of the proof

Now let us replace g by its Langlands dual Lie algebra Lg. Then we obtainthe following commutative diagram

π∨0 (Lg)ν0

∼−−−−→ FunConn(Ωρ)D,ν0x xW(Lg)ν0

∼−−−−→ FunOpLG(D)ν0

(8.2.12)

of Poisson vertex algebras, which is compatible with the action of DerO andAutO.

This completes the sixth, and last, step of our plan from Section 7.1. Wehave now assembled all the pieces needed to describe the center z(g) of Vκc(g).

Combining the commutative diagrams (8.2.12) and (8.1.10) and taking intoaccount Theorem 8.2.4, Theorem 8.1.7 and Proposition 8.1.8 we come to the

† We remark that the screening operators, understood as derivations of π∨0 (g) or

FunConn(Ωρ)D, do not commute with the action of DerO. However, we can makethem commute with DerO if we consider them as operators acting from π∨0 (g) to an-other module, isomorphic to π∨0 (g) as a vector space, but with a modified action of DerO.Since we will not use this fact here, we refer a curious reader to [FF7] for more details.


following result (here by LG we understand the group of inner automorphismsof Lg).

Theorem 8.2.5 There is an isomorphism z(g)κ0 ' FunOpLG(D)κ∨0which

preserves the vertex Poisson structures and the DerO-module structures onboth sides. Moreover, it fits into a commutative diagram of vertex Poissonalgebras equipped with DerO-action:

π0(g)κ0

∼−−−−→ FunConn(Ωρ)D,κ∨0x xz(g)κ0

∼−−−−→ FunOpLG(D)κ∨0

(8.2.13)

where the upper arrow is given on the generators by the assignment bi,n 7→−ui,n.

In particular, we have now proved Theorem 4.3.2.Note that Theorem 8.2.5 is consistent with the description of π0(g) given

previously in Section 6.2.5. According to this description, the bi(t)’s transformunder the action of DerO as components of a connection on the LH-bundleΩ−ρ on the disc D = Spec C[[t]]. Therefore π0(g) is identified, in a DerO-equivariant way, with the algebra FunConn(Ωρ)D of functions on the spaceof connections on Ω−ρ. Thus, the top isomorphism in the diagram (8.2.13)may be expressed as an isomorphism

FunConn(Ω−ρ)D ' FunConn(Ωρ)D. (8.2.14)

Under this isomorphism a connection ∂t + u(t) on Ωρ is mapped to the dualconnection ∂t − u(t) on the dual LH-bundle Ω−ρ. This precisely correspondsto the map bi,n 7→ −ui,n on the generators of the two algebras, which appearsin Theorem 8.2.5.

Given any disc Dx, we may consider the twists of our algebras by the AutO-torsor Autx, defined as in Section 6.2.6. We will mark them by the subscriptx. Then we obtain that

W(g)ν0,x ' FunOp(Dx)ν0 ,

z(g)κ0,x ' FunOpLG(Dx)κ∨0.

8.2.6 The associated graded algebras

In this section we describe the isomorphism z(g) ' FunOpLG(D) of Theo-rem 8.2.5 at the level of associated graded spaces. We start by describing thefiltrations on z(g) and FunOpLG(D).

The filtration on z(g) is induced by the Poincare–Birkhoff–Witt filtration on


the universal enveloping algebra U(gκc), see Section 3.3.3. By Theorem 8.1.5,gr z(g) = (grVκc

(g))g[[t]]. But

grVκc(g) = Sym g((t))/g[[t]] ' Fun g∗[[t]]dt,

independently of the choice of coordinate t and inner product on g. InProposition 3.4.2 we gave a description of (grVκc(g))g[[t]]. The coordinate-independent version of this description is as follows. Let Cg = Spec (Fun g∗)G

and

Cg,Ω = Ω ×C×

Cg,

where Ω = C[[t]]dt is the topological module of differentials onD = Spec C[[t]].By Proposition 3.4.2 we have a canonical and coordinate-independent isomor-phism

gr z(g) ' FunCg,Ω. (8.2.15)

Note that a choice of homogeneous generators P i, i = 1, . . . , `, of (Fun g∗)G

gives us an identification

Cg,Ω '⊕i=1

Ω⊗(di+1),

but we prefer not to use it as there is no natural choice for such generators.Next we consider the map

a : z(g)→ π0(g)

which is equal to the restriction of the embedding Vκc(g)→W0,κc to z(g). Inthe proof of Proposition 7.1.1 we described a filtration on W0,κc

compatiblewith the PBW filtration on Vκc

(g). This implies that the map z(g) → π0(g)is also compatible with filtrations. According to the results of Section 6.2.5,we have a canonical identification

π0(g) = FunConn(Ω−ρ)D.

The space Conn(Ω−ρ)D of connections on the LH-bundle Ω−ρ is an affinespace over the vector space Lh⊗ Ω = h∗ ⊗ Ω. Therefore we find that

grπ0(g) = Fun h∗ ⊗ Ω. (8.2.16)

We have the Harish-Chandra isomorphism (Fun g∗)G ' (Fun h∗)W , whereW is the Weyl group of g, and hence an embedding (Fun g∗)G → Fun h∗. Thisembedding gives rise to an embedding

FunCg,Ω → Fun h∗ ⊗ Ω.

It follows from the proof of Proposition 7.1.1 that this is precisely the map

gr a : gr z(g)→ grπ0(g)


under the identifications (8.2.15) and (8.2.16). Thus, we have now describedthe associated graded of the left vertical map in the commutative diagram(8.2.13).

Next, we describe the associated graded of the right vertical map in thecommutative diagram (8.2.13). Let

C∨g = g/G = Spec(Fun g)G

and

C∨g,Ω = Ω ×C×

C∨g .

Following [BD1], Section 3.1.14, we identify OpG(D) with an affine spacemodeled on C∨g,Ω. In other words, we have a natural filtration on FunOpG(D)such that

grOpG(D) = FunC∨g,Ω. (8.2.17)

Constructing such a filtration is the same as constructing a flat C[h]-algebrasuch that its specialization at h = 1 is FunOpG(D), and the specialization ath = 0 is FunC∨g,Ω. Consider the algebra of functions on the space OpG,h(X) ofh-opers onD. The definition of an h-oper is the same as that of an oper, exceptthat we consider instead of a connection of the form (4.2.1) an h-connection

h∂t +∑i=1

ψi(t)fi + v(t).

One shows that this algebra is flat over C[h] by proving that each h-oper hasa canonical form

h∂t + p−1 + v(t), v(t) ∈ Vcan((t)),

in exactly the same way as in the proof of Lemma 4.2.2.It is clear that OpG,1(D) = OpG(D). In order to see that OpG,0(D) = C∨g,Ω,

observe that it follows from the definition that OpG,0(D) = Γ(D,Ω ×C×

Vcan),

where the action of C× on Vcan is given by a 7→ aAdρ(a). But it follows from[Ko] that we have a canonical isomorphism of C×-spaces Vcan ' g/G. Thisimplies that OpG,0(D) = C∨g,Ω and hence (8.2.17).

Now consider the space of connections Conn(Ωρ)D on the LH-bundle Ωρ.It is an affine space over the vector space Lh⊗ Ω. Therefore

gr Fun Conn(Ωρ)D = Fun Lh⊗ Ω. (8.2.18)

We have the Harish-Chandra isomorphism (Fun g)G ' (Fun h)W , where Wis the Weyl group of g, and hence an embedding (Fun g)G → Fun h. Thisembedding gives rise to an embedding

FunC∨g,Ω → Fun h⊗ Ω.

8.3 The center of the completed universal enveloping algebra 237

The explicit construction of the Miura transformation Conn(Ωρ)D → OpLG(D)given in the proof of Proposition 8.2.2 implies that the map

FunOpLG(D)→ FunConn(Ωρ)D

preserves filtrations. Furthermore, with respect to the identifications (8.2.17)and (8.2.18), its associated graded map is nothing but the homomorphism

FunC∨Lg,Ω → Fun Lh⊗ Ω.

This describes the associated graded of the right vertical map in the commu-tative diagram (8.2.13).

Next, the associated graded of the upper horizontal isomorphism is givenby the natural composition

grπ0(g) ' Fun h∗ ⊗ Ω = Fun Lh⊗ Ω ' gr Fun Conn(Ωρ)D,

multiplied by the operator (−1)deg which takes the value (−1)n on elementsof degree n (this is due to the map bi,n 7→ −ui,n in Theorem 8.2.5).

Note that we have canonical isomorphisms

(Fun g∗)G = (Fun h∗)W = (Fun Lh)W = (Fun Lg)LG,

which give rise to a canonical identification

FunCg,Ω = FunC∨Lg,Ω.

Now, all maps in the diagram (8.2.13) preserve the filtrations. The commuta-tivity of the diagram implies that the associated graded of the lower horizontalisomorphism

z(g) ' FunOpLG(D) (8.2.19)

is equal to the composition

gr z(g) ' FunCg,Ω = FunC∨Lg,Ω ' gr Fun OpLG(D)

multiplied by the operator (−1)deg which takes the value (−1)n on elementsof degree n. Thus, we obtain the following:

Theorem 8.2.6 The isomorphism (8.2.19) preserves filtrations. The corre-sponding associated graded algebras are both isomorphic to FunCg,Ω. The cor-responding isomorphism of the associated graded algebras is equal to (−1)deg.

8.3 The center of the completed universal enveloping algebra

Recall the completion Uκ(g) of the universal enveloping algebra of gκ definedin Section 2.1.2. Let Z(g) be its center. In Theorem 4.3.6 we have derived from


Theorem 4.3.2 (which we have now proved, see Theorem 8.2.5) the followingAutO-equivariant isomorphism:

Z(g) ' FunOpLG(D×). (8.3.1)

In this section we discuss various properties of this isomorphism.

8.3.1 Isomorphism between Z(g) and FunOpLG(D×)

Recall the AutO-equivariant isomorphism of vertex Poisson algebras

z(g)κ0 ' FunOpLG(D)κ∨0(8.3.2)

established in Theorem 8.2.5. For a vertex Poisson algebra P , the Lie algebrastructure on U(P ) = Lie(P ) gives rise to a Poisson algebra structure on thecommutative algebra U(P ). In particular, U(z(g)κ0) = Z(g) (see Proposi-tion 4.3.4) is a Poisson algebra.

The corresponding Poisson structure on Z(g)κ0 may be described as fol-lows. Consider Uκ(g) as the one-parameter family Aε of associative algebrasdepending on the parameter ε = (κ− κc)/κ0. Then Z(g) is the center of A0.Given x, y ∈ Z(g), let x, y be their liftings to Aε. Then the Poisson bracketx, y is defined as the ε-linear term in the commutator [x, y] considered as afunction of ε (it is independent of the choice of the liftings).† We denote thecenter Z(g) equipped with this Poisson structure by Z(g)κ0 .

Likewise, the vertex Poisson algebra FunOpLG(D)κ∨0gives rise to a topo-

logical Poisson algebra U(FunOpLG(D)κ∨0). According to Lemma 4.3.5, the

latter is isomorphic to FunOpLG(D×) in an AutO-equivariant way. Thereforethe isomorphism (8.3.2) gives rise to an isomorphism of topological Poissonalgebras

Z(g)κ0 ' FunOpLG(D×)κ∨0.

Here we use the subscript κ∨0 to indicate the dependence of the Poisson struc-ture on the inner product κ∨0 on Lg. We will give another definition of thisPoisson structure in the next section. Thus, we obtain the following result,which was originally conjectured by V. Drinfeld.

Theorem 8.3.1 The center Z(g)κ0 is isomorphic, as a Poisson algebra, tothe Poisson algebra FunOpLG(D×)κ∨0

. Moreover, this isomorphism is AutO-equivariant.

† The fact that this is indeed a Poisson structure was first observed by V. Drinfeld followingthe work [Ha] of T. Hayashi.


8.3.2 The Poisson structure on FunOpG(D×)ν0

We have obtained above a Poisson structure on the algebra Fun OpG(D×)from the vertex Poisson structure on FunOpG(D), which was in turn obtainedby realizing Fun OpG(D) as a vertex Poisson subalgebra of π∨(g)ν0 (namely, asthe classical W-algebra W(g)ν0). Now we explain how to obtain this Poissonstructure by using the Hamiltonian reduction called the Drinfeld–Sokolovreduction [DS].

We start with the Poisson manifold Conng of connections on the trivial G-bundle on D×, i.e., operators of the form ∇ = ∂t + A(t), where A(t) ∈ g((t)).The Poisson structure on this manifold comes from its identification with ahyperplane in the dual space to the affine Kac–Moody algebra gν0 , where ν0 isa non-zero invariant inner product on g. Indeed, the topological dual space togν0 may be identified with the space of all λ-connections on the trivial bundleon D×, see [FB], Section 16.4. Namely, we split gν0 = g((t))⊕ C1 as a vectorspace. Then a λ-connection ∂t + A(t) gives rise to a linear functional on gν0

which takes the value

λb+ Res ν0(A(t), B(t))dt

on (B(t) + b1) ∈ g((t)) ⊕ C1 = gν0 . Note that under this identification theaction of AutO by changes of coordinates and the coadjoint (resp., gauge)action of G((t)) on the space of λ-connections (resp., the dual space to gν0)agree. The space Conng is now identified with the hyperplane in g∗ν0

whichconsists of those functionals which take the value 1 on the central element 1.

The dual space g∗ν0carries a canonical Poisson structure called the Kirillov-

Kostant structure (see, e.g., [FB], Section 16.4.1 for more details). Because 1is a central element, this Poisson structure restricts to the hyperplane whichwe have identified with Conng. Therefore we obtain a Poisson structure onConng.

The group N((t)) acts on Conng by gauge transformations. This action cor-responds to the coadjoint action of N((t)) on g∗ν0

and is Hamiltonian, themoment map being the surjection m : Conng → n((t))∗ dual to the em-bedding n((t)) → gν0 . We pick a one-point coadjoint N((t))-orbit in n((t))∗

represented by the linear functional ψ which is equal to the compositionn((t))→ n/[n, n]((t)) =

⊕ì=1 C((t)) · ei and the functional

(xi(t))ì=1 7→

∑i=1

Rest=0 xi(t)dt.

One shows in the same way as in the proof of Lemma 4.2.2 that the action ofN((t)) on m−1(ψ) is free. Moreover, the quotient m−1(ψ)/N((t)), which is thePoisson reduced manifold, is canonically identified with the space of G-operson D×. Therefore we obtain a Poisson structure on the topological algebra offunctions FunOpG(D×).


Thus, we now have two Poisson structures on Fun OpG(D×) associated toa non-zero invariant inner product ν0 on g.

Lemma 8.3.2 The two Poisson structures coincide.

Proof. In [FB], Sections 15.4 and 16.8, we defined a complex C•∞(g)and showed that its zeroth cohomology is canonically isomorphic to W(g)ν0 ,equipped with the vertex Poisson algebra structure introduced in Section 8.1.4(and all other cohomologies vanish). This complex is a vertex Poisson algebraversion of the BRST complex computing the result of the Drinfeld–Sokolovreduction described above. In particular, we identify FunOpG(D×), equippedwith the Poisson structure obtained via the Drinfeld–Sokolov reduction, withU(W(g)ν0), with the Poisson structure corresponding to the above vertexPoisson structure on W(g)ν0 .

8.3.3 The Miura transformation as the Harish-Chandra

homomorphism

The Harish-Chandra homomorphism, which we have already discussed in Sec-tion 5.1, is a homomorphism from the center Z(g) of U(g), where g is a simpleLie algebra, to the algebra Fun h∗ of polynomials on h∗. It identifies Z(g) withthe algebra (Fun h∗)W of W -invariant polynomials on h∗. To construct thishomomorphism, one needs to assign a central character to each λ ∈ h∗. Thiscentral character is just the character with which the center acts on the Vermamodule Mλ−ρ.

In the affine case, we construct a similar homomorphism from the centerZ(g) of Uκc(g) to the topological algebra FunConn(Ω−ρ)D× of functions onthe space of connections on the LH-bundle Ω−ρ on D×. According to Corol-lary 6.1.5, points of Conn(Ω−ρ)D× parameterize Wakimoto modules of criticallevel. Thus, for each ∇ ∈ Conn(Ω−ρ)D× we have the Wakimoto module W∇of critical level. The following theorem describes the affine analogue of theHarish-Chandra homomorphism.

Note that FunConn(Ω−ρ)D× is the completion of the polynomial algebrain bi,n, i = 1, . . . , `;n ∈ Z, with respect to the topology in which the base ofopen neighborhoods of 0 is formed by the ideals generated by bi,n, n < N . Inthe same way as in the proof of Lemma 4.3.5 we show that it is isomorphic toU(FunConn(Ω−ρ)D). We also define the topological algebra FunConn(Ωρ)D×

as U(FunConn(Ωρ)D). It is the completion of the polynomial algebra inui,n, i = 1, . . . , `;n ∈ Z, with respect to the topology in which the base ofopen neighborhoods of 0 is formed by the ideals generated by ui,n, n < N . Theisomorphism (8.2.14) gives rise to an isomorphism of the topological algebras

FunConn(Ω−ρ)D× → FunConn(Ωρ)D× ,


under which bi,n 7→ −ui,n.The natural forgetful morphism MOpLG(D×)gen → OpLG(D×) and an

identification

MOpLG(D×)gen ' Conn(Ωρ)D×

constructed in the same way as in Proposition 8.2.2 give rise to a map

µ : Conn(Ωρ)D× → OpLG(D×). (8.3.3)

This is the Miura transformation on the punctured disc D×, which is analo-gous to the Miura transformation (8.2.3) on the disc D.

We recall the general construction of semi-infinite parabolic induction de-scribed in Corollary 6.3.2. We apply it in the case of the Borel subalgebrab ⊂ g. Let M be a module over the vertex algebra Mg, or, equivalently, amodule over the Weyl algebra Ag. Let R be a module over the commutativevertex algebra π0 = FunConn(Ω−ρ)D, or, equivalently, a smooth module overthe commutative algebra Fun Conn(Ω−ρ)D× . Then the homomorphism

wκc: Vκc

(g)→Mg ⊗ π0

of Theorem 6.1.6 gives rise to the structure of a Vκc(g)-module, or, equiva-lently, a smooth gκc

–module on the tensor product Mg ⊗R.Now we obtain from Theorem 8.2.5 the following result.

Theorem 8.3.3 The action of Z(g) on the gκc-module M ⊗ R does not

depend on M and factors through a homomorphism

Z(g)→ FunConn(Ω−ρ)D×

and the action of FunConn(Ω−ρ)D× on R. The corresponding morphismConn(Ω−ρ)D× → SpecZ(g) fits into a commutative diagram

Conn(Ω−ρ)D×∼−−−−→ Conn(Ωρ)D×y y

SpecZ(g) ∼−−−−→ OpLG(D×)

(8.3.4)

where the right vertical arrow is the Miura transformation (8.3.3) on D×.In particular, if R = C∇ is the one-dimensional module corresponding to

a point ∇ ∈ Conn(Ω−ρ)D× , then the center Z(g) acts on the correspondingWakimoto module W∇ = Mg ⊗ C∇ via a central character corresponding tothe Miura transformation of ∇.

Proof. The action of U(Vκc(g)), and hence of Uκc

(g), on W∇ is obtainedthrough the homomorphism of vertex algebras Vκc(g) → Mg ⊗ π0(g). Inparticular, the action of Z(g) on W∇ is obtained through the homomorphismof commutative vertex algebras z(g)→ π0(g). But π0(g) = FunConn(Ω−ρ)D,according to Section 6.2.5. Therefore the statement of the theorem follows


from Theorem 8.2.5 by applying the functor of enveloping algebras V 7→ U(V )introduced in Section 3.2.3.

Thus, we see that the affine analogue of the Harish-Chandra homomorphismis the right vertical arrow of the diagram (8.3.4), which is nothing but theMiura transformation for the Langlands dual group! In particular, its image(which in the finite-dimensional case consists of W -invariant polynomials onh∗) is described as the intersection of the kernels of the W-algebra screeningoperators.

9

Structure of g-modules of critical level

In the previous chapters we have described the center z(g) of the vertex al-gebra Vκc

(g) at the critical level. As we have shown in Section 3.3.2, z(g) isisomorphic to the algebra of endomorphisms of the gκc

-module Vκc(g), which

commute with the action of gκc. We have identified this algebra with the

algebra of functions on the space OpLG(D) of LG-opers on the disc D.In this chapter we obtain similar results about the algebras of endomor-

phisms of the Verma modules and the Weyl modules of critical level. Weshow that both are quotients of the center Z(g) of the completed envelopingalgebra Uκc

(g), which is isomorphic to the algebra of functions on the spaceOpLG(D) of LG-opers on the punctured disc D×. In the case of Verma mod-ules, the algebra of endomorphisms is identified with the algebra of functionson the space of opers with regular singularities and fixed residue, and in thecase of Weyl modules it is the algebra of functions of opers with regular sin-gularities, fixed residue and trivial monodromy. Thus, the geometry of opersis reflected in the representation theory of affine algebras of critical level. Un-derstanding this connection between representation theory and geometry isimportant for the development of the local Langlands correspondence that wewill discuss in the next chapter.

We begin this chapter by introducing the relevant subspaces of the spaceOpLG(D×): opers with regular singularities (in Section 9.1) and nilpotentopers (in Section 9.2), and explain the interrelations between them. We thenconsider in Section 9.3 the restriction of the Miura transformation definedin Section 8.2.1 to opers with regular singularities and nilpotent opers. Werelate the fibers of the Miura transformation over the nilpotent opers to theSpringer fibers. This will allow us to obtain families of Wakimoto modulesparameterized by the Springer fibers. Next, we discuss in Section 9.4 somecategories of representations of gκc

that “live” over the spaces of opers withregular singularities. Finally, in Sections 9.5 and 9.6 we describe the algebrasof endomorphisms of the Verma modules and the Weyl modules of criticallevel.

The results of Section 9.1 are due to A. Beilinson and V. Drinfeld [BD1],

243

244 Structure of g-modules of critical level

and most of the results of the remaining sections of this chapter were obtainedby D. Gaitsgory and myself in [FG2, FG6].

9.1 Opers with regular singularity

In this section we introduce, following Beilinson and Drinfeld [BD1], the spaceof opers on the disc with regular singularities at the origin.

9.1.1 Definition

Recall that the space OpG(D) (resp., OpG(D×)) of G-opers on D (resp., D×)is the quotient of the space of operators of the form (4.2.1) where ψi(t) andv(t) take values in C[[t]] (resp., in C((t))) by the action of B[[t]] (resp., B((t))).

A G-oper on D with regular singularity is by definition (see [BD1],Section 3.8.8) a B[[t]]-conjugacy class of operators of the form

∇ = ∂t + t−1

(∑i=1

ψi(t)fi + v(t)

), (9.1.1)

where ψi(t) ∈ C[[t]], ψi(0) 6= 0, and v(t) ∈ b[[t]].Equivalently, it is an N [[t]]-equivalence class of operators

∇ = ∂t +1t

(p−1 + v(t)) , v(t) ∈ b[[t]]. (9.1.2)

Denote by OpRSG (D) the space of opers on D with regular singularity.

More generally, we consider, following [BD1], Section 3.8.8, the spaceOpordk

G (D) of opers with singularity of order less than or equal to k > 0as the space of N [[t]]-equivalence classes of operators

∇ = ∂t +1tk

(p−1 + v(t)) , v(t) ∈ b[[t]].

9.1.2 Residue

Following [BD1], we associate to an oper with regular singularity its residue.For an operator (9.1.2) the residue is by definition equal to p−1+v(0). Clearly,under gauge transformations by an element x(t) of N [[t]] the residue getsconjugated by x(0) ∈ N . Therefore its projection onto

g/G = Spec(Fun g)G = Spec(Fun h)W = h/W

is well-defined. Thus, we obtain a morphism

res : OpRSG (D)→ h/W.

Given $ ∈ h/W , we denote by OpRSG (D)$ the space of opers with regular

singularity and residue $.

9.1 Opers with regular singularity 245

9.1.3 Canonical representatives

Suppose that we are given an oper with regular singularity represented by aB[[t]]-conjugacy class of a connection ∇ of the form (9.1.1). Recall that theset E = d1, . . . , d` is the set of exponents of g counted with multiplicityand the subspace V can =

⊕i∈E V

cani of g with a basis p1, . . . , p` introduced

in Section 4.2.4.In the same way as in the proof of Lemma 4.2.2 we use the gauge action of

B[[t]] to bring ∇ to the form

∂t + t−1

p−1 +∑j∈E

vj(t)pj

, vj(t) ∈ C[[t]]. (9.1.3)

The residue of this operator is equal to

p−1 +∑j∈E

vj(0)pj . (9.1.4)

Denote by gcan the affine subspace of g consisting of all elements of the formp−1 +

∑j∈E yjpj . Recall from [Ko] that the adjoint orbit of any regular

elements in the Lie algebra g contains a unique element which belongs to gcan

and the corresponding morphism gcan → h/W is an isomorphism. Thus, weobtain that when we bring an oper with regular singularity to the canonicalform (9.1.3), its residue is realized as an element of gcan.

Consider the natural morphism OpRSG (D) → OpG(D×) taking the B[[t]]-

equivalence class of operators of the form (9.1.1) to its B((t))-equivalence class.

Proposition 9.1.1 ([DS], Prop. 3.8.9) The map OpRSG (D) → OpG(D×)

is injective. Its image consists of those G-opers on D× whose canonical rep-resentatives have the form

∂t + p−1 +∑j∈E

t−j−1cj(t)pj , cj(t) ∈ C[[t]]. (9.1.5)

Moreover, the residue of this oper is equal to

p−1 +(c1(0) +

14

)p1 +

∑j∈E,j>1

cj(0)pj . (9.1.6)

Proof. First, we bring an oper with regular singularity to the form (9.1.3).Next, we apply the gauge transformation by ρ(t)−1 and obtain the following

∂t + p−1 + ρt−1 +∑j∈E

t−j−1vj(t)pj , vj(t) ∈ C[[t]].

Finally, applying the gauge transformation by exp(−p1/2t) we obtain the


operator

∂t + p−1 + t−2

(v1(t)−

14

)p1 +

∑j∈E,j>1

t−j−1vj(t)pj , vj(t) ∈ C[[t]].

(9.1.7)Thus, we obtain an isomorphism between the space of opers with regular

singularity and the space of opers onD× of the form (9.1.5), and, in particular,we find that the map OpRS

G (D)→ OpG(D×) is injective. Moreover, comparingformula (9.1.7) with formula (9.1.5), we find that c1(t) = v1(t)− 1

4 and cj(t) =vj(t) for j > 1. By (9.1.4), the residue of the oper given by formula (9.1.5) isequal to (9.1.6).

As a corollary, we obtain that for any point $ in h/W ' gcan the operswith regular singularity and residue $ form an affine subspace of OpRS

G (D).Using the isomorphism (4.2.9), we can phrase Proposition 9.1.1 in a coordi-

nate-independent way. Denote by Proj≥−2 the space of projective connectionson D× having pole of order at most two at the origin and by ω⊗(dj+1)

≥−(dj+1) thespace of (dj + 1)-differentials on D× having pole of order at most (dj + 1) atthe origin.

Corollary 9.1.2 There is an isomorphism between OpRSG (D) and the space

Proj≥−2 ×⊕j>1

ω⊗(dj+1)

≥−(dj+1).

Under this isomorphism, OpRSG (D)$ corresponds to the affine subspace con-

sisting of the `-tuples (η1, . . . , η`) satisfying the following condition. Denoteby ηi,−di−1 the t−di−1-coefficient of the expansion of ηi in powers of t (it isindependent of the choice of t). Then the conjugacy class in g/G = h/W of

p−1 +(η1,−2 +

14

)p1 +

∑j>1

ηj,−j−1pj

is equal to $.

Suppose that we are given a regular oper

∂t + p−1 + v(t), v(t) ∈ b[[t]].

Using the gauge transformation with ρ(t) ∈ B((t)), we bring it to the form

∂t +1t

(p−1 − ρ+ t · ρ(t)(v(t))ρ(t)−1

).

If v(t) is regular, then so is ρ(t)v(t)ρ(t)−1. Therefore this oper has the form(9.1.2), and its residue is equal to $(−ρ), where $ is the projection h→ h/W .Since the map OpRS

G (D) → OpG(D×) is injective (by Proposition 9.1.1), wefind that the (affine) space OpLG(D) is naturally realized as a subspace of thespace of opers with regular singularity and residue $(−ρ).

9.2 Nilpotent opers 247

9.2 Nilpotent opers

Now we introduce another realization of opers with regular singularity whoseresidue has the form $(−λ − ρ), where λ is the dominant integral weight,i.e., such that 〈αi, λ〉 ∈ Z+, and $(−λ− ρ) is the projection of (−λ− ρ) ∈ h

onto h/W . This realization will be convenient for us because it reveals a“secondary” residue invariant of an oper, which is an element of n/B (asopposed to the “primary” residue, defined above, which is an element of h/W ).

To illustrate the difference between the two realizations, consider the caseof g = sl2. As before, we identify h with C by sending ρ 7→ 1. An sl2-operwith residue −1− λ, where λ ∈ Z+, is the N [[t]]-gauge equivalence class of anoperator of the form

∂t +

(− 1+λ

2t + a(t) b(t)1t

1+λ2t − a(t)

),

where a(t), b(t) ∈ C[[t]]. This operator may be brought to the canonicalform by applying gauge transformations as in the proof of Proposition 9.1.1.Namely, we apply the gauge transformation with(

1 − 12t

0 1

)(t−1/2 0

0 t1/2

)(1 1+λ

2 − ta(t)0 1

)to obtain the operator in the canonical form

∂t +

(0 1

t2

(1+λ

2 − ta(t))2

− 14t2 + 1

t (b(t) + a(t)) + a′(t)

1 0

).

On the other hand, if we apply the gauge transformation with the matrix

(ρ+ λ)(t)−1 =

(t(−1−λ)/2 0

0 t(1+λ)/2

),

then we obtain the following operator:

∂t +

(a(t) b(t)

tλ+1

tλ −a(t)

).

Let b(t) =∑

n≥0 bntn. Then, applying the gauge transformation with(

1∑λ−1

n=0bn

−λ−nt−λ−n

0 1

),

we obtain an operator of the form

∂t +

(a(t)

eb(t)t

tλ −a(t)

),

where a(t), b(t) ∈ C[[t]].


This is an example of a nilpotent oper. Its residue, as a connection (not tobe confused with the residue as an oper!), is the nilpotent matrix(

0 b00 0

),

which is well-defined up to B-conjugation. From this we find that the mon-odromy of this oper is conjugate to the matrix(

1 2πib00 1

).

Thus, in addition to the canonical form, we obtain another realization ofopers with integral residue. The advantage of this realization is that we havea natural residue map and so we can see more clearly what the monodromicproperties of these opers are.

In this section we explain how to generalize this construction to other simpleLie algebras, following [FG2].

9.2.1 Definition

Let λ ∈ h be a dominant integral coweight. Consider the space of operatorsof the form

∇ = ∂t +∑i=1

t〈αi,λ〉ψi(t)fi + v(t) +v

t, (9.2.1)

where ψi(t) ∈ C[[t]], ψi(0) 6= 0, v(t) ∈ b[[t]] and v ∈ n. The group B[[t]] actson this space by gauge transformations. Since B[[t]] is a subgroup of the groupB((t)) which acts freely on the space of all operators of them form (4.2.1) (seeSection 4.2.4), it follows that this action of B[[t]] is free. We define the spaceOpnilp,λ

G of nilpotent G-opers of coweight λ as the quotient of the spaceof all operators of the form (9.2.1) by B[[t]].

We have a map

βλ : Opnilp,λG → OpG(D×)

taking the B[[t]]-equivalence class of operator ∇ of the form (9.2.1) to itsB((t))-equivalence class. This map factors through a map

δλ : Opnilp,λG → OpRS

G (D)$(−λ−ρ),

where $(−λ − ρ) is the projection of (−λ − ρ) ∈ h onto h/W , which isconstructed as follows.


Given an operator ∇ of the form (9.2.1), consider the operator

(λ+ ρ)(t)∇(λ+ ρ)(t)−1 = ∂t + t−1

(∑i=1

ψi(t)fi + w(t)

),

w(t) ∈ (λ+ ρ)(t)b[[t]](λ+ ρ)(t)−1, w(0) = −λ− ρ. (9.2.2)

Moreover, under this conjugation the B[[t]]-equivalence class of ∇ maps tothe equivalence class of (λ+ ρ)(t)∇(λ+ ρ)(t)−1 with respect to the subgroup

(λ+ ρ)(t)B[[t]](λ+ ρ)(t)−1 ⊂ B[[t]].

Now, the desired map δλ assigns to the B[[t]]-equivalence class of ∇ theB[[t]]-equivalence class of (λ+ ρ)(t)∇(λ+ ρ)(t)−1. By construction, the latteris an oper with regular singularity and residue $(−λ− ρ).

9.2.2 Nilpotent opers and opers with regular singularity

Clearly, the map βλ is the composition of δλ and the inclusion

OpRSG (D)$(−λ−ρ) → OpG(D×).

Proposition 9.2.1 The map δλ is an isomorphism for any dominant integralcoweight λ.

Proof. We will show that for any C-algebra R the morphism δλ givesrise to a bijection at the level of R-points. This would imply that δλ is anisomorphism not only set-theoretically, but as a morphism of schemes.

Thus, we need to show that any operator of the form (9.1.1) with residue$(−λ − ρ) may be brought to the form (9.2.2) over any R, and moreover,that if any two operators of the form (9.2.2) are conjugate under the gaugeaction of g ∈ B(R[[t]]), then g ∈ (λ+ ρ)(t)B(R[[t]])(λ+ ρ)(t)−1.

We will use the notation

µ = λ+ ρ =∑

i

µiωi.

By our assumption on µ, we have µi ≥ 1 for all i.Note that

(λ+ ρ)(t)B(R[[t]])(λ+ ρ)(t)−1 = H(R[[t]]) · (λ+ ρ)(t)N(R[[t]])(λ+ ρ)(t)−1.

There is a unique element ofH(R[[t]]) that makes each function ψi(t) in (9.1.1)equal to 1. Hence we need to show that any operator ∇ of the form (9.1.2)with residue $(−λ− ρ) may be brought to the form

∇ = ∂t + t−1 (p−1 + v(t)) , v(t) ∈ µ(t)b[[t]]µ(t)−1, v(0) = −µ, (9.2.3)

under the action of N(R[[t]]), and, moreover, that if any two operators ofthis form are conjugate under the gauge action of g ∈ N(R[[t]]), then g ∈


µ(t)N(R[[t]])µ(t)−1. In order to simplify notation, we will suppress R andwrite N [[t]] for N(R[[t]]), etc.

Consider first the case when λ = 0, i.e., when µi = 1 for all i. We will bringan operator (9.1.2) to the form (9.2.3) by induction. We have a decomposition

ρ(t)b[[t]]ρ(t)−1 = h[[t]]⊕h⊕

j=1

tjnj [[t]],

where n =⊕

j>0 nj is the decomposition of n with respect to the principalgradation and h is the Coxeter number (see Section 4.2.4). We will assumethat h > 1, for otherwise g = sl2 and the operator is already in the desiredform.

Recall that, by our assumption, the residue of ∇ is equal to $(−ρ). Hencewe can use the gauge action of the constant subgroup N ⊂ N [[t]] to bring ∇to the form

∂t +1t

(p−1 − ρ) + v(t), (9.2.4)

where v(t) ∈ b[[t]]. Let is decompose v(t) =∑h

j=0 vj(t) with respect to theprincipal gradation and apply the gauge transformation by

g = exp(− 1h− 1

tvh(0))∈ N [[t]].

The constant coefficient of the principal degree h term in the new operatorreads

vh(0) +1

h− 1[−vh(0),−ρ] +

1h− 1

vh(0) = 0

(the second term comes from −g ρtg−1 and the last term from g∂t(g−1)). Thus,

we find that in the new operator vnew,h(t) ∈ tnh[[t]].If h = 2, then we are done. Otherwise, we apply the gauge transformation

by exp(− 1h−2 tvh−1(0)). (Here and below we denote by vi(t) the components

of the operator obtained after the latest gauge transformation.) We obtain inthe same way that the new operator will have vi(t) ∈ tni[[t]] for i = h− 1, h.Continuing this way, we obtain an operator of the form (9.2.4) in which v1(t) ∈n1[[t]] and vj(t) ∈ tnj [[t]] for j = 2, . . . , h. (It is easy to see that each of thesegauge transformations preserves the vanishing conditions that were obtainedat the previous steps.)

Next, we wish to obtain an operator of the form (9.2.4), where v1(t) ∈n1[[t]],v2(t) ∈ tC[[t]] and vj(t) ∈ t2nj [[t]] for j = 3, . . . , h. This is done in thesame way as above, by consecutively applying the gauge transformations with

exp(− 1j − 2

t2v′j(0)), j = h, h− 1, . . . , 3,


where v′j(0) denotes the t-coefficient in the jth component of the operatorobtained at the previous step. Clearly, this procedure works over an arbitraryC-algebra R.

Proceeding like this, we bring our operator to the form

∂t +1t

(p−1 + v(t)) , v(t) ∈ ρ(t)b[[t]]ρ(t)−1, v(0) = −ρ, (9.2.5)

which is (9.2.3) with µ = ρ. Now let us show that if ∇ and g∇g−1 are ofthe form (9.2.5) and g ∈ N [[t]], then g necessarily belongs to the subgroupρ(t)N [[t]]ρ(t)−1 (this is equivalent to saying that the intersection of the N [[t]]-equivalence class of the initial operator ∇ with the set of all operators of theform (9.2.5), where µ = ρ, is a ρ(t)N [[t]]ρ(t)−1-equivalence class).

Since the gauge action of any element of the constant subgroup N ⊂ N [[t]]other than the identity will change the residue of the connection, we mayassume without loss of generality that g belongs to the first congruence sub-group N (1)[[t]] of N [[t]]. Since the exponential map exp : tn[[t]] → N (1)[[t]]is an isomorphism, we may represent any element of N (1)[[t]] uniquely as theproduct of elements of the form exp(cα,nt

n), where cα,n ∈ nα, in any cho-sen order. In particular, we may write any element of N (1)[[t]] as the prod-uct of two elements, one of which belongs to the subgroup ρ(t)N [[t]]ρ(t)−1,and the other is the product of elements of the form exp(cα,nt

n), whereα ∈ ∆+, 1 ≤ n < 〈α, ρ〉. But we have used precisely the gauge transfor-mation of an element of the second type to bring our operator to the form(9.2.5) and they were fixed uniquely by this process. Therefore applying any ofthem to the operator (9.2.5) will change its form. So the subgroup of N (1)[[t]]consisting of those transformations that preserve the form (9.2.5) is preciselythe subgroup µ(t)N [[t]]µ(t)−1.

This completes the proof in the case where λ = 0, µ = ρ.We will prove the statement of the proposition for a general µ by induction.

Observe first that the same argument as above shows that any operator ofthe form (9.1.1) with residue $(µ) may be brought to the form

∂t + t−1 (p−1 + v(t)) , v(t) ∈ ρ(t)b[[t]]ρ(t)−1, v(0) = −µ,

for any µ = λ+ ρ, where λ is a dominant integral coweight. Let us show thatthis operator may be further brought to the form

∂t + t−1 (p−1 + v(t)) , v(t) ∈ µ′(t)b[[t]]µ′(t)−1, v(0) = −µ, (9.2.6)

where µ′ is any dominant integral coweight such that

1 ≤ 〈αi, µ′〉 ≤ 〈αi, µ〉, i = 1, . . . , `. (9.2.7)

We already know that this is true for µ′ = ρ. Suppose we have proved itfor some µ′. Let us prove it for the coweight µ + ωi, assuming that it stillsatisfies condition (9.2.7). So we have an operator of the form (9.2.6), and


we wish to bring it to the same form where µ′ is replaced by µ′ + ωi. Usingthe root decomposition n =

⊕α∈∆+

nα, we write v(t) =∑

α∈∆+vα(t), where

vα(t) ∈ nα. We have

vα(t) =∑

n≥〈α,µ′〉

vα,ntn. (9.2.8)

We will proceed by induction. At the first step we consider the componentsvα(t), where α is a simple root (i.e., those which belong to n1). We have〈α, µ′ + ωi〉 = 〈α, µ′〉 for all such α’s, except for α = αi, for which we have〈α, µ′ + ωi〉 = 〈α, µ′〉+ 1.

Let us write µi = 〈αi, µ〉, µ′i = 〈αi, µ′〉. Then condition (9.2.7) for µ′ + ωi

will become

2 ≤ µ′i + 1 ≤ µi. (9.2.9)

Apply the gauge transformation by

exp(

1µ′i − µi

tµ′ivαi,µ′i

)(note that the denominator is non-zero due condition (9.2.9)). As the result,we obtain an operator of the form (9.2.6) where vαi

(t) ∈ tµ′i+1C[[t]]. At thesame time, we do not spoil condition (9.2.8) for any other α ∈ ∆+, because〈α, µ′〉 ≥ 〈α, ρ〉 by our assumption on µ′.

Next, we move on to roots of height two (i.e., those which belong to n2).Let us apply consecutively the gauge transformations by

exp(

1〈α, µ′〉 − 〈α, µ〉

t〈α,µ′〉vα,〈α,µ′〉

)for those α’s for which 〈α, µ′ + ωi〉 > 〈α, µ′〉. Continuing this process acrossall values of the principal gradation, we obtain an operator which satisfiescondition (9.2.8) for all α’s for which 〈α, µ′〉 = 〈α, µ′ + ωi〉 and the condition

vα(t) =∑

n≥〈α,µ′〉+1

vα,ntn

for all α’s for which 〈α, µ′ + ωi〉 > 〈α, µ′〉.At the next step we obtain in a similar way an operator which satisfies a

stronger condition, namely, that in addition to the above we have

vα(t) =∑

n≥〈α,µ′〉+2

vα,ntn

for those α’s for which 〈α, µ′ + ωi〉 > 〈α, µ′〉 + 1. Condinuing this way, wefinally arrive at an operator for which

vα(t) =∑

n≥〈α,µ′+ωi〉

vα,ntn,


for all α ∈ ∆+. In other words, it has the form (9.2.3) where µ′ is replacedby µ′ + ωi. This completes the inductive step.

Therefore we obtain that any operator of the form (9.1.2) with residue$(−µ) may be brought to the form (9.2.3) (clearly, this construction worksover any C-algebra R). Finally, we prove in the same way as in the case whenµ = ρ that if any two operators of this form are conjugate under the gaugeaction of g ∈ N [[t]], then g ∈ µ(t)N [[t]]µ(t)−1.

By Proposition 9.1.1 and Proposition 9.2.1, the map βλ : Opnilp,λG →

OpG(D×) is an injection. In particular, using Proposition 9.1.1 we obtaincanonical representatives of nilpotent opers. In what follows we will not dis-tinguish between Opnilp,λ

G and its image in OpG(D×).

9.2.3 Residue

Let FB be the B-bundle on the discD underlying an oper χ ∈ Opnilp,λG , and let

FB,0 be its fiber at 0 ∈ D. One can show in the same way as in Section 4.2.4that the B-bundles FB underlying all λ-nilpotent opers on D are canonicallyidentified. Denote by nFB,0 the FB,0-twist of n. Then for each operator ofthe form (9.2.1), v is a well-defined vector in nFB,0 . Therefore we obtain amorphism

ResF,λ : Opnilp,λG → nFB,0 , χ 7→ v.

For each choice of trivialization of FB,0 we obtain an identification nFB,0 ' n.Changes of the trivialization lead to the adjoint action of B on n. Thereforewe have a canonical map nFB,0 → n/B. Its composition with ResF,λ is amorphism

Resλ : Opnilp,λG → n/B.

It maps an operator of the form (9.2.1) to the projection of v onto n/B. Wecall it the residue morphism. Note that any nilpotent G-oper, viewed as aconnection on D×, has a monodromy operator, which is a conjugacy class inG. The following statement is clear.

Lemma 9.2.2 Any nilpotent G-oper χ of coweight λ has unipotent mon-odromy. Its monodromy is trivial if and only if Resλ(χ) = 0.

Let OpλG be the preimage of 0 ∈ n/B in Opnilp,λ

G under the map Resλ.Equivalently, this is the space of B[[t]]-equivalence classes of operators of theform

∇ = ∂t +∑i=1

t〈αi,λ〉ψi(t)fi + v(t), (9.2.10)


where ψi(t) ∈ C[[t]], ψi(0) 6= 0, v(t) ∈ b[[t]]. This space was originally intro-duced by V. Drinfeld (unpublished).

The above lemma and Proposition 9.2.1 imply that OpλG is the submanifold

in OpRSG (D)$(−λ−ρ) corresponding to those opers with regular singularity that

have trivial monodromy.Consider the case when λ = 0 in more detail. We will denote Opnilp,0

G

simply by OpnilpG and call its points nilpotent G-opers. These are the B[[t]]-

equivalence classes of operators of the form

∂t +∑i=1

ψi(t)fi + v(t) +v

t, (9.2.11)

where ψi(t) ∈ C[[t]], ψi(0) 6= 0, v(t) ∈ b[[t]] and v ∈ n, or, equivalently, theN [[t]]-equivalence classes of operators of the form

∂t + p−1 + v(t) +v

t, (9.2.12)

where v(t) ∈ b[[t]] and v ∈ n. The residue morphism, which we denote in thiscase simply by Res, maps such an operator to the image of v in n/B. Notethat the fiber over 0 ∈ n/B is just the locus Op0

G = OpG(D) of regular operson the disc D.

Proposition 9.1.1 identifies OpnilpG with the preimage of $(−ρ) under the

map res : OpRSG (D)→ h/W defined in Section 9.1.1. Using Proposition 9.2.1

and Proposition 9.1.1, we obtain the following description of OpnilpG .

Corollary 9.2.3 There is an isomorphism

OpnilpG ' Proj≥−1 ×

⊕j≥2

ω⊗(dj+1)≥−dj

. (9.2.13)

9.2.4 More general forms for opers with regular singularity

The argument we used in the proof of Proposition 9.2.1 allows us to constructmore general representatives for opers with regular singularities.

Consider the space OpRS(D)$(−µ), where µ is an arbitrary element of h.An element of OpRS(D)$(−µ) is the same as an N (1)[[t]]-equivalence class ofoperators of the form

∂t +1t(p−1 − µ) + v(t), v(t) ∈ b[[t]]. (9.2.14)

Let λ′ be a dominant integral coweight such that

〈α, λ′〉 < 〈α, µ〉, if 〈α, µ〉 ∈ Z>0, α ∈ ∆+, (9.2.15)

where Z>0 is the set of positive integers.Set µ′ = λ′+ρ. Let Op(D)µ′

µ be the space of gauge equivalence classes of the


operators of the form (9.2.6) by the gauge action of the group µ′(t)N [[t]]µ′(t)−1.Then we have a natural map

δµ′

µ : Op(D)µ′

µ → OpRS(D)$(−µ)

taking these equivalence classes to the N [[t]]-equivalence classes. Applyingverbatim the argument used in the proof of Proposition 9.2.1, we obtain thefollowing result.

Proposition 9.2.4 If λ′ satisfies condition (9.2.15), and µ′ = λ′ + ρ, thenthe map δµ′

µ is an isomorphism.

In particular, suppose that 〈α, µ〉 6 ∈Z>0 for all α ∈ ∆+. Note that anypoint in h/W may be represented as $(−µ) where µ satisfies this condition.Then the statement of Proposition 9.2.4 is valid for any dominant integralweight λ′. Let us set λ′ = nρ, n ∈ Z+. Then we have µ′ = (n + 1)ρ andµ′(t)N [[t]]µ′(t)−1 is the (n + 1)st congruence subgroup of N [[t]]. Taking thelimit n→∞, we obtain the following:

Corollary 9.2.5 Suppose that µ satisfies the property that 〈α, µ〉 6∈Z>0 forall α ∈ ∆+. Then any oper with regular singularity and residue $(−µ) maybe written uniquely in the form

∂t +1t(p−1 − µ) + v(t), v(t) ∈ h[[t]],

or, equivalently, in the form

∂t + p−1 −µ+ ρ

t+ v(t), v(t) ∈ h[[t]]. (9.2.16)

Thus, there is a bijection between OpRS(D)$(−µ) and the space of operatorsof the form (9.2.16).

9.2.5 A characterization of nilpotent opers using the Deligne

extension

Recall that given a G-bundle F with a connection ∇ on the punctured discD× which has unipotent monodromy, there is a canonical extension of F ,defined by P. Deligne [De1], to a G-bundle F on the disc D such that theconnection has regular singularity and its residue is nilpotent. Consider aG-oper (F ,FB ,∇) on D×. Suppose that it has unipotent monodromy. AnyG-bundle on D× with a connection whose monodromy is unipotent has acanonical extension to a bundle on D with a connection with regular singu-larity and nilpotent residue, called the Deligne extension. Thus, our bundleF on D× has the Deligne extension to D, which we denote by F .

Since the flag variety is proper, the B-reduction FB of F on D× extends


to a B-reduction of F on D, which we denote by FB . We refer to (F ,FB ,∇)as the Deligne extension of the oper (F ,FB ,∇) to D. By construction, theconnection ∇ has regular singularity and its residue is an element of

gFB,0= FB ×

FB

g,

where FB,0 is the fiber of FB at the origin.Recall the H-torsor O which is an open dense subset in n⊥/b ⊂ g/b. Sup-

pose we are given a function f on D with values in n⊥/b which takes valuesin O over D×. Given an integral coweight λ, we will say that f vanishes toorder λ at the origin if the corresponding map D× → O may be obtained asthe restriction of the cocharacter λ : C× → O under an embedding D× → C×and an identification O ' H. Note that this definition still makes sense ifinstead of a function with values in n⊥/b we take a section of a twist of n⊥/b

by an H-bundle on D.We will say that (F ,FB ,∇) satisfies the λ-oper condition at the origin if

the one-form ∇/FB on D taking values in (n⊥/b)FB(and in OFB

⊂ (g/b)FB

over D×) vanishes to order λ at the origin. The following statement followsdirectly from the definitions.

Proposition 9.2.6 Let λ be an integral dominant coweight. A G-oper on D×

is a nilpotent G-oper of coweight λ if and only if it has unipotent monodromyand its Deligne extension satisfies the λ-oper condition at the origin.

9.3 Miura opers with regular singularities

In Section 8.2.1 we introduced the spaces of Miura opers and Cartan connec-tions and the Miura transformation to the space of opers. In this section weconsider Miura opers and Cartan connections with regular singularities andthe restriction of the Miura transformation to the space of Cartan connectionswith regular singularities. We show that the fiber of the Miura transformationover a nilpotent oper χ is isomorphic to the Springer fiber of the residue of χ.The results of this section were obtained in [FG2].

9.3.1 Connections and opers with regular singularities

In Section 8.2.1 we defined the space Conn(Ωρ)D of connections on the H-bundle Ωρ on the discD = Spec C[[t]]. Now we consider the space Conn(Ωρ)D×

of connections on this bundle on the punctured disc D× = Spec C((t)). Ele-ments of Conn(Ωρ)D× are represented by operators of the form

∇ = ∂t + u(t), u(t) ∈ h((t)), (9.3.1)

and under changes of coordinates they transform according to formula (8.2.2).

9.3 Miura opers with regular singularities 257

Now let Conn(Ωρ)RSD be the space of all connections on the H-bundle Ωρ

on D with regular singularity, that is, of the form

∇ = ∂t +1tu(t), u(t) ∈ h[[t]]. (9.3.2)

We have a map

resh : Conn(Ωρ)RSD → h

assigning to a connection its residue u(0) ∈ h. For λ ∈ h let Conn(Ωρ)λD be

the subspace of connections with residue λ.Recall that we have the Miura transformation (8.3.3)

µ : Conn(Ωρ)D× → OpG(D×) (9.3.3)

defined as follows: given an operator ∇ of the form (9.3.1), we define thecorresponding G-oper as the N((t))-gauge equivalence class of the operator∇+ p−1.

Suppose that∇ is an element of Conn(Ωρ)RSD given by formula (9.3.2). Then

µ(∇) is by definition the N((t))-equivalence class of the operator

∇ = ∂t + p−1 + t−1u(t).

Applying gauge transformation with ρ(t), we identify it with the N((t))-equivalence class of the operator

∂t + t−1(p−1 − ρ+ u(t)),

which is an oper with regular singularity and residue $(−ρ+ u(0)).Thus, we obtain a morphism

µRS : Conn(Ωρ)RSD → OpRS

G (D). (9.3.4)

and a commutative diagram

Conn(Ωρ)RSD

µRS

−−−−→ OpRSG (D)

resh

y yres

h −−−−→ h/W

(9.3.5)

where the lower horizontal map is the composition of the map λ 7→ λ− ρ andthe projection $ : h→ h/W .

It is easy to see that the preimage of any χ ∈ OpRSG (D) under the Miura

transformation (9.3.3) belongs to Conn(Ωρ)RSD ⊂ Conn(Ωρ)D× . In other

words, the oper corresponding to a Cartan connection with irregular singu-larity necessarily has irregular singularity. Thus, µRS is the restriction of µ toOpRS

G (D). The above commutative diagram then implies that for any λ ∈ h


we have an isomorphism†

µ−1(OpRSG (D)$(−λ−ρ)) =

⊔w∈W

Conn(Ωρ)ρ−w(λ+ρ)D . (9.3.6)

In addition, Corollary 9.2.5 implies the following:

Proposition 9.3.1 If λ ∈ h is such that 〈α, λ〉 6∈Z+ for all α ∈ ∆+, thenthe restriction of the map µRS to Conn(Ωρ)−λ

D is an isomorphism

Conn(Ωρ)−λD ' OpRS

G (D)$(−λ−ρ).

Thus, if λ satisfies the conditions of the Proposition 9.3.1, then the stratumcorresponding to w = 1 in the right hand side of (9.3.6) gives us a section ofthe restriction of the Miura transformation µ to OpRS

G (D)$(−λ−ρ)).The following construction will be useful in what follows. For an H-bundleFH on D and an integral coweight λ, let FH(λ) = FH(λ · 0), where 0 denotesthe origin in D (the closed point of D), be the H-bundle on D such that forany character µ : H → C× we have

FH(λ) ×C×

Cµ = (FH × Cµ)(〈µ, λ〉 · 0).

Note that the restrictions of the bundles FH(λ) and FH to D× are canonicallyidentified. Under this identification, a connection ∇ on FH(λ) becomes theconnection λ(t)−1∇λ(t) on FH .

Denote by Conn(Ωρ(λ))D the space of connections on the H-bundle Ωρ(λ)on D. We have a natural embedding

Conn(Ωρ(λ))D ⊂ Conn(Ωρ)D×

obtained by restricting a connection to D×. The image of Conn(Ωρ(λ))D

in Conn(Ωρ)D× coincides with the space Conn(Ωρ)λD ⊂ Conn(Ωρ)D× . In-

deed, given a connection ∇ ∈ Conn(Ωρ(λ)D, the connection λ(t)−1∇λ(t) is inConn(Ωρ)λ

D.In what follows we will use the notation Conn(Ωρ)λ

D for both of these spaceskeeping in mind the above two distinct realizations.

9.3.2 The case of integral dominant coweights

Let λ be an integral dominant coweight of G. A nilpotent Miura G-oper ofcoweight λ on the disc D is by definition a quadruple (F ,∇,FB ,F ′B), where(F ,∇,FB) is a nilpotent G-oper of coweight λ on D and F ′B is another B-reduction of F stable under ∇t∂t

= t∇. Let MOpλG be the variety of nilpotent

Miura G-opers of coweight λ.

† Here and throughout this chapter we consider the set-theoretic preimage of µ. Thecorresponding scheme-theoretic preimage may be non-reduced (see examples in Sec-tion 10.4.6).


The restriction of a nilpotent Miura G-oper of coweight λ from D to D× isa Miura oper on D×, as defined in Section 8.2.1. Let MOpG(D×) be the setof Miura opers on D×. We have a natural forgetful map MOpλ

G → Opnilp,λG

which fits into the following Cartesian diagram:

MOpλG −−−−→ MOpG(D×)y y

Opnilp,λG −−−−→ OpG(D×)

The proof of the next lemma is due to V. Drinfeld (see [FG2], Lemma 3.2.1).Recall from Section 8.2.1 that a Miura oper (F ,∇,FB ,F ′B) is called genericif the B-reductions FB and F ′B are in generic relative position.

Lemma 9.3.2 Any Miura oper on the punctured disc D× is generic.

Proof. Let us choose a coordinate t on the disc D and trivialize the B-bundle FB . Then we identify the sections of F over D× with G((t)), thereduction F ′B with a right coset g ·B((t)) ⊂ G((t)) and the connection ∇ withthe operator

∂t +A(t), A(t) =∑i=1

ψi(t)fi + v(t), (9.3.7)

where ψi(t) ∈ C((t))× and v(t) ∈ b[[t]].We have the Bruhat decomposition G((t)) =

⊔w∈W B((t))wB((t)). Therefore

we may write g = g1wg2 for some w ∈W and g1, g2 ∈ B((t)). Hence the spaceof sections over D× of the Borel subalgebra bF ′B ⊂ gF ′B = gFB

is equal to

b′ = gb((t))g−1 = g1(wb((t))w−1)g−11 .

The statement of the lemma is equivalent to the statement that the elementw appearing in this formula is the longest element w0 of the Weyl group W .

To see that, consider the Cartan subalgebra h′ = g1h((t))g−11 . Then, since

g1 ∈ B((t)), we have h′ ⊂ b′ ∩ b((t)). We have the following Cartan decompo-sition of the Lie algebra g((t)) with respect to h′:

g((t)) = h′ ⊕⊕α∈∆

g′α, g′α = g1gαg−11 ,

where gα is the root subspace of g corresponding to α ∈ ∆. Let us decomposeA(t) appearing in formula (9.3.7) with respect to this decomposition:

A(t) = A0 +∑α∈∆

Aα.

Then because of the oper condition we have A−αi 6= 0. On the other hand,since F ′B is preserved by ∇, we have A(t) ∈ b′. This means that the roots−αi are positive roots with respect to b′. Therefore all negative roots with


respect to b((t)) are positive roots with respect to b′. This is possible if andonly if b′ = g1b−g

−11 , where b− = w0bw

−10 and g1 ∈ B((t)).

In the same way as in Lemma 8.2.2 we show that there is a bijection betweenthe set of generic Miura opers on D× and the set Conn(Ωρ)D× of connectionson the H-bundle Ωρ on D×. Then Lemma 9.3.2 implies:

Lemma 9.3.3 There is a bijection

MOpG(D×) ' Conn(Ωρ)D× .

Thus, any nilpotent Miura oper becomes generic, when restricted to thepunctured disc, and hence defines a connection on Ωρ. Our goal now is tounderstand the relationship between the residue of this connection and therelative position between the B-reductions FB and F ′B at 0 ∈ D.

Let N ⊂ g be the cone of nilpotent elements in g and N the Springervariety of pairs (x, b′), where x ∈ N and b′ is a Borel subalgebra of g thatcontains x. Let n be the restriction of N to n ⊂ N . Thus,

n = (x, b′) |x ∈ n, x ∈ b′.

Recall that in Section 9.2.3 we defined the residue Resλ(χ) of a nilpotentoper χ of coweight λ.

Lemma 9.3.4 The reduction F ′B of a nilpotent oper χ of coweight λ isuniquely determined by its fiber at the origin. It must be stable under theresidue of the underlying nilpotent oper, Resλ(χ).

To see that, recall that F ′B may be completely described by choosing linesubbundles Lν in the vector bundle V ν

F = F ×GV ν , where V ν is an irreducible

representation of G of highest weight ν, satisfying the Plucker relations (see,e.g., [FGV1], Section 2.1.2). Then the above statement is a corollary of thefollowing:

Lemma 9.3.5 Suppose we are given a first order differential equation withregular singularity

(∂t +A(t))Ψ(t) = 0, A(t) =∑

i≥−1

Aiti,

on a finite-dimensional vector space V . Suppose that A−1 is a nilpotent op-erator on V . Then this equation has a solution Ψ(t) ∈ V [[t]] with the initialcondition Ψ(0) = v ∈ V if and only if A−1(v) = 0.

Proof. We expand Ψ(t) =∑j≥0

Ψjtj . Then the above equation is equivalent


to the system of linear equations

nΨn +∑

i+j=n−1

Ai(Ψj) = 0. (9.3.8)

The first of these equations reads A−1(Ψ0) = 0, so this is indeed a necessarycondition. If this condition is satisfied, then the Ψn’s with n > 0 may befound recursively, because the nilpotency of A−1 implies that the operator(n Id+A−1), n > 0, is invertible.

This implies that we have a natural isomorphism

MOpλG ' Opnilp,λ

G ×n/B

n/B, (9.3.9)

where the map Opnilp,λG → n/B is the residue map Resλ. Thus, the fiber of

MOpλG over χ ∈ Opnilp,λ

G is the (reduced) Springer fiber of Resλ(χ):

SpResλ(χ) = b′ ∈ G/B | Resλ(χ) ∈ b′. (9.3.10)

Given w ∈ W , we will say that a Borel subgroup B′ (or the correspondingLie algebra b′) is in relative position w with B (resp., b) if the point of theflag variety G/B corresponding to B′ (resp., b′) lies in the B-orbit Bw−1w0B.Denote by Nw the subvariety of N which consists of those pairs (x, b′) forwhich b′ is in relative position w with our fixed Borel subalgebra b. Let nw

be the restriction of Nw to n, respectively. The group B naturally acts on n

preserving nw.Let MOpλ,w

G be the subvariety of MOpλG consisting of those Miura opers for

which the fiber of F ′B at the origin is in relative position w with B. Then wehave an identification

MOpλ,wG = Opnilp,λ

G ×n/B

nw/B. (9.3.11)

9.3.3 Connections and Miura opers

According to Proposition 9.2.1, for any integral dominant coweight λ we havean isomorphism

OpRSG (D)−$(λ+ρ) ' Opnilp,λ

G .

Hence the restriction of the morphism µRS (see formula (9.3.4)) to the sub-space Conn(Ωρ)−w(λ+ρ)+ρ

D ⊂ Conn(Ωρ)RSD gives us a morphism

µ−w(λ+ρ)+ρ : Conn(Ωρ)−w(λ+ρ)+ρD → Opnilp,λ

G .

Now, it follows from the explicit construction of the Miura transformationµ (see Section 9.3.1) that for any nilpotent oper χ of coweight λ, which is inthe image of the map µ−w(λ+ρ)+ρ, the Borel reduction

F ′B = Ωρ ×Hw0B


is stable under t∇. Therefore the map µ−w(λ+ρ)+ρ may be canonically liftedto a map

µ−w(λ+ρ)+ρ : Conn(Ωρ)−w(λ+ρ)+ρD → MOpλ

G . (9.3.12)

This map fits into the Cartesian diagram

Conn(Ωρ)−w(λ+ρ)+ρD −−−−→ Conn(Ωρ)D×

eµ−w(λ+ρ)+ρ

y yMOpλ

G −−−−→ MOpG(D×)

where the right vertical map is the bijection established in Lemma 9.3.3. Thefollowing result is proved in [FG2], Cor. 3.6.5 (for λ = 0).

Theorem 9.3.6 The map µ−w(λ+ρ)+ρ is an isomorphism

Conn(Ωρ)−w(λ+ρ)+ρD ' MOpλ,w

G ⊂ MOpλG .

Proof. Suppose that we are given a point of MOpλG, i.e., a nilpotent

oper of coweight λ equipped with a Borel reduction F ′B that is stable undert∇. Since our nilpotent Miura oper is generic on D×, by Lemma 9.3.2, itfollows from Lemma 8.2.1 that the H-bundle F ′B/N is isomorphic over D×

to w∗0(FB/N) = w∗0(Ωρ). Thus, we obtain a connection on w∗0(Ωρ) and hencea connection on Ωρ over D×. Denote the latter by ∇. It is clear that ∇ hasregular singularity, and so

∇ = ∂t −1t(µ+ t(. . .)), µ ∈ h.

Clearly, the Miura oper (F ,∇,FB ,F ′B) onD× obtained from∇ by applyingthe bijection of Lemma 9.3.3, is the Miura oper that we have started with.Therefore the corresponding oper has the form

∂t +1t

(p−1 − ρ− µ+ t(. . .)) .

Hence its residue in h/W is equal to $(−µ− ρ). But by our assumption theresidue of this oper is equal to $(−λ − ρ). Therefore we obtain that thereexists y ∈W such that −ρ− µ = y(−λ− ρ), i.e., µ = y(λ+ ρ)− ρ.

Thus, we obtain that the restriction of the bijection of Lemma 9.3.3 toMOpλ

G is a bijection

MOpλG '

⊔y∈W

Conn(Ωρ)−y(λ+ρ)+ρD .

We need to show that under this bijection MOpw,λG corresponds precisely to

Conn(Ωρ)−w(λ+ρ)+ρD .

Recall that we have an isomorphism of theH-bundles FH |D× ' w∗0(F ′H)|D×


over D×, where FH = FB/N,F ′H = F ′B/N . Let us observe that if we havetwo H-bundles FH and F ′′H over D, then any isomorphism FH |D× ' F ′′H |D×

extends to an isomorphism FH ' F ′′H(µ) over D, for some integral coweight µ.By considering the transformation properties of a nilpotent oper of coweight λunder changes of variables in the same way as as in the proof of Lemma 4.2.1,we find that

FH = FB/N ' Ωρ(−λ).

The assertion of the theorem will follow if we show that our isomorphismFH |D× ' w∗0(F ′H)|D× extends to an isomorphism

F ′H ' w∗0(FH((λ+ ρ)− w(λ+ ρ)) (9.3.13)

over D, because then

F ′H ' w∗0(Ωρ(ρ− w(λ+ ρ)),

and so the induced connection on Ωρ on D× is in the image of

Conn(Ωρ(ρ− w(λ+ ρ))D ' Conn(Ωρ)ρ−w(λ+ρ)D ,

as desired.Let us trivialize the fiber of F at the origin in such a way that FB gets

identified with B ⊂ G. Without loss of generality, we may assume that underthis trivialization F ′B,0 becomes w−1w0B. We need to show the following. Fora dominant integral weight ν let V ν be the finite-dimensional representationof g of highest weight ν. For w ∈ W we pick a non-zero vector vw ∈ V ν inthe one-dimensional subspace of V ν of weight w(ν). By our assumption, F ′Bis preserved by t∇. Therefore by Lemma 9.3.4, the residue of ∇ stabilizes thevector vw−1w0 . Consider the associated vector bundle V ν

F = F ×GV ν with the

connection induced by ∇. It follows from Lemma 9.3.5 that there is a uniquesingle-valued horizontal section of this flat bundle whose value at t = 0 is equalto vw−1w0 . We express Ψ(t) as the sum of components of different weights.Let φw(t)vw0 , where φw(t) ∈ C[[t]], be the component of Ψ(t) of the lowestweight w0(ν). By Lemma 9.3.2, F ′B and FB are in generic position over D×.This implies that φw(t) 6= 0. Let nw be the order of vanishing of φw(t) att = 0.

We need to show that for each dominant integral weight ν we have

nw = 〈w0(ν), w(λ+ ρ)− (λ+ ρ)〉, (9.3.14)

because this is equivalent to formula (9.3.13).To compute the order of vanishing of φw(t), recall that the connection

underlying a nilpotent oper of coweight λ has the form

∇ = ∂t +∑i=1

t〈αi,λ〉fi + v(t) +v

t, (9.3.15)


where v(t) ∈ b[[t]], v ∈ n. The sought-after horizontal section of V νF may

be obtained by solving the equation ∇Ψ(t) = 0 with values in V ν [[t]]. Wesolve this equation recursively, as in the proof of Lemma 9.3.5. Then wehave to solve equations (9.3.8), in which on the right hand side we have alinear combination of operators in b, which preserve or increase the principalgradation, and the operators fi, i = 1, . . . , `, which lower it by 1. It is thereforeclear that the leading term of the component φw(t)vw0 of Ψ(t) proportionalto vw0 is obtained by applying the operators fi, i = 1, . . . , `, to vw−1w0 . Henceit is the same as for the connection

∇ = ∂t +∑i=1

t〈αi,λ〉ψi(t)fi.

When we solve the recurrence relations (9.3.8) for this connection, each appli-cation of fi leads to the increase the t-degree of Ψ(t) by 〈αi, λ〉+1 = 〈αi, λ+ρ〉.This is the “(λ+ ρ)-degree” of αi. Therefore to “reach” vw0 from vw−1w0 (theinitial condition for Ψ(t)) we need to increase the t-degree by the differencebetween the (λ+ ρ)-degrees of the vectors vw−1w0 and vw0 , i.e., by

〈w−1w0(ν), λ+ ρ〉 − 〈w0(ν), λ+ ρ〉 = 〈w0(ν), w(λ+ ρ)− (λ+ ρ)〉,

which is nw given by formula (9.3.14). Hence we find that φw(t) = cwtnw

for some cw ∈ C. If cw were equal to 0, then the entire vw0-componentof the solution Ψ(t) would be equal to 0. But this would mean that thecorresponding Borel reductions F ′B and FB are not in generic position overD×. This contradicts Lemma 9.3.2. Hence cw 6= 0, and so for any oper(9.3.15) the leading t-degree of φw(t) is indeed equal to nw. This completesthe proof.

Combining the isomorphisms (9.3.12) and (9.3.11), we obtain an isomor-phism

Conn(Ωρ)ρ−w(λ+ρ)D ' Opnilp,λ

G ×n/B

nw/B. (9.3.16)

Thus, we have a bijection of the sets of C-points⊔w∈W

Conn(Ωρ)ρ−w(λ+ρ)D ' Opnilp,λ

G ×n/B

n/B. (9.3.17)

While (9.3.16) is an isomorphism of varieties, (9.3.17) is not, because the lefthand side is a disjoint union of strata, which are “glued” together in the righthand side.

Recall that according to formula (9.3.6), the left hand side of (9.3.17) is pre-cisely the preimage of Opnilp,λ

G ⊂ OpG(D×) under the Miura transformationµ. Thus, we see that under the isomorphism (9.3.17) the natural projection

Opnilp,λG ×

n/Bn/B → Opnilp,λ

G

9.4 Categories of representations of g at the critical level 265

on the first factor coincides with the restriction of the Miura transformationµ (see (9.3.3)) to Opnilp,λ

G . This implies that the set of points in the fiberµ−1(χ) of the Miura transformation µ over χ ∈ Opnilp,λ

G is in bijection withthe set of points of the Springer fiber SpResλ(χ) (see (9.3.10)).

As in the case of the isomorphism (9.3.17), this is only a bijection at thelevel of C-points. The fiber µ−1(χ) is the union of strata

µ−1(χ)w = µ−1(χ) ∩ Conn(Ωρ)ρ−w(λ+ρ)D , w ∈W,

and each of these strata is isomorphic, as a variety, to the subvariety

SpwResλ(χ) = SpResλ(χ) ∩Sw,

consisting of those Borel subalgebras that are in relative position w with b;here Sw is the Schubert cell in G/B corresponding to w (see the proof ofLemma 9.3.2).

We summarize this in the following statement.

Theorem 9.3.7 For any χ ∈ Opnilp,λG , the fiber µ−1(χ) of the Miura trans-

formation µ over χ is the union of strata

µ−1(χ)w ⊂ Conn(Ωρ)ρ−w(λ+ρ)D , w ∈W.

The stratum µ−1(χ)w is isomorphic to

SpwResλ(χ) = SpResλ(χ) ∩Sw,

where SpResλ(χ) is the Springer fiber of Resλ(χ). Hence the set of points ofµ−1(χ) is in bijection with the set of points of SpResλ(χ).

For example, if Resλ(χ) = 0, then SpResλ(χ) is the flag variety G/B andSpw

Resλ(χ) is the Schubert cell Sw. Thus, we find that µ−1(χ) is in bijectionwith the set of points of G/B in this case.

Note that nw0 is a single point (namely, b′ = b), and so µ−1(χ)w0 =Spw0

Resλ(χ) consists of a single point. Therefore we have an isomorphism

Conn(Ωρ)ρ−w0(λ+ρ)D ' Opnilp,λ

G . (9.3.18)

This is actually a special case of the statement of Proposition 9.3.1 (seealso Corollary 9.2.5), because if λ is an integral dominant coweight then thecoweight µ = w0(λ+ ρ)− ρ satisfies 〈α, µ〉 6∈Z+ for all α ∈ ∆+.

9.4 Categories of representations of g at the critical level

In the previous sections we defined certain subvarieties of the space OpG(D×)of G-opers on the punctured disc D×. Let us now switch from G to theLanglands dual group LG (which we will assume to be of adjoint type, asbefore). In particular, this means that we switch from the space Conn(Ωρ)D×


of connections on the H-bundle Ωρ and its subspaces defined above to thespace Conn(Ωρ) of connections on the LH-bundle Ωρ and the correspondingsubspaces.

According to Theorem 8.3.1, the space OpLG(D×) is isomorphic to thespectrum of the center Z(g) of the completed enveloping algebra of g of criticallevel. Thus, the category of all smooth gκc

-modules “lives” over OpLG(D×).In this section we identify certain subcategories of this category, which “live”over the subspaces OpRS

LG(D)$ ⊂ OpLG(D×).

9.4.1 Compatibility with the finite-dimensional Harish-Chandra

homomorphism

We start by describing a certain compatibility between the Miura transfor-mation and the Harish-Chandra isomorphism Z(g) ' (Fun h∗)W , where Z(g)be the center of U(g)

Consider the center Z(g) of Uκc(g). We have a natural Z-gradation on Z(g)

by the operator L0 = −t∂t ∈ DerO.Let us denote by the subscript 0 the degree 0 part and by the subscript < 0

the span of all elements of negative degrees in any Z-graded object.Next, consider the Wakimoto modules Wχ(t), where

χ(t) =λ− ρt

, λ ∈ h∗

(see the proof of Theorem 6.4.1). These Wakimoto modules are Z-graded, withthe degree 0 part (Wχ(t))0 being the subspace C[a∗α,0|0〉]α∈∆+ ⊂Mg ⊂Wχ(t).The Lie algebra g preserves this subspace, and it follows immediately from ourconstruction of Wakimoto modules that as a g-module, (Wχ(t))0 is isomorphicto the contragredient Verma module M∗λ−ρ. The algebra

Z ′ = Z(g)0/(Z(g) · Z(g)<0)0

naturally acts on (Wχ(t))0 and commutes with g. Moreover, since its actionnecessarily factors through the action of U(g), we find that it factors throughZ(g). Therefore varying λ ∈ h∗ we obtain a homomorphism

Z ′ → (Fun h∗)W , (9.4.1)

which factors through the Harish-Chandra homomorphism Z(g)→ (Fun h∗)W .On the other hand, recall from Section 9.1.1 the space

OpRSLG(D) ⊂ OpLG(D×)

of LG-opers with regular singularity and its subspace OpRSLG(D×)$ of opers

with residue $ ∈ Lh/W = h∗/W (recall that h∗ = Lh).We have a Z-gradation on FunOpLG(D×) induced by the operator L0 =


−t∂t ∈ DerO. If we write the oper connection in the canonical form ofLemma 4.2.2,

∇ = ∂t + p−1 +∑j=1

vj(t) · pj , vj(t) =∑n∈Z

vj,nt−n−1,

then we obtain from the transformation formulas of Section 4.2.4 that

L0 · vj,n = (−n+ dj)vj,n.

Therefore the generators vj,n are homogeneous, and those of strictly nega-tive degrees are vj,n, n > dj , which by Corollary 9.1.2 generate the ideal ofOpRS

G (D) in FunOpG(D×). Therefore the ideal of OpRSG (D) is generated by

FunOpLG(D×)<0. Clearly, the residue map

res : OpRSG (D)→ h∗/W

gives rise to an isomorphism

(Fun h∗)W ' (FunOpRSLG(D))0.

We also have the space

Conn(Ωρ)RSD ⊂ Conn(Ωρ)D×

of connections with regular singularity on the LH-bundle Ωρ on D× definedin Section 9.3.1. The Miura transformation restricts to a map (see (9.3.4))

µRS : Conn(Ωρ)RSD → OpRS

LG(D).

We have a Z-gradation on Fun Conn(Ωρ)D× induced by the operator L0 =−t∂t ∈ DerO. If we write ∇ ∈ Conn(Ωρ)D× in the form ∇ = ∂t +u(t), where

ui(t) = αi(u(t)) =∑n<0

ui,nt−n−1,

then we have

L0 · ui,n = −nui,n.

Therefore the generators ui,n are homogeneous, and those of strictly neg-ative degrees are vi,n, n > 0. Those generate the ideal of Conn(Ωρ)RS

D inFunConn(Ωρ)D× . The residue map

resLh : OpRSG (D)→ h∗

defined in Section 9.3.1 gives rise to an isomorphism

Fun h∗ → (FunConn(Ωρ)RSD )0.


We have a commutative diagram (9.3.5)

Conn(Ωρ)RSD

µRS

−−−−→ OpRSLG(D)

resLh

y yres

h∗ −−−−→ h∗/W

It gives rise to a commutative diagram of algebra homomorphisms

Fun h∗∼−−−−→ (FunConn(Ωρ)RS

D )0x x(Fun h∗)W ∼−−−−→ (FunOpRS

LG(D))0

(9.4.2)

Now, since the ideal of OpRSLG(D) in FunOpLG(D×) is generated by the sub-

algebra FunOpLG(D×)<0 we obtain that the isomorphism of Theorem 8.3.1gives rise to an isomorphism of algebras

Z ′ ' (FunOpLG(D×)/(FunOpLG(D×)<0))0 = (Fun OpRSLG(D))0. (9.4.3)

Therefore commutative diagram (9.4.2) implies that the map (9.4.1) is anisomorphism.

Recall that in the isomorphism of Theorem 8.3.3 we have bi,n 7→ −ui,n.Therefore we obtain the following:

Proposition 9.4.1 ([F4],Prop. 12.8) There is a commutative diagram ofisomorphisms

Z(g) ∼−−−−→ (FunOpRSLG(D))0y x

(Fun h∗)W ∼−−−−→ (Fun h∗)W

where the lower horizontal arrow is given by f 7→ f−, f−(λ) = f(−λ).

9.4.2 Induction functor

We define a functor from the category of g-modules to the category of gκc -modules on which 1 acts as the identity. Let M be a g-module. Define theinduced gκc -module Ind(M) as

Ind(M) = Indbgκc

g[[t]]⊕1M,

where g[[t]] acts on M through the evaluation homomorphism g[[t]] → g and1 acts as the identity.

In particular, if M = Mλ, the Verma module of highest weight λ ∈ h∗, thenthe induced module Ind(Mλ) is the Verma module over gκc of critical level,denoted by Mλ,κc , which we have encountered previously.


Let again Z(g) be the center of U(g). The Harish-Chandra homomorphismZ(g) ∼→ Fun(h∗)W gives rise to an isomorphism h∗/W → SpecZ(g). Noweach λ ∈ h∗ defines a point $(λ) ∈ h∗/W and hence a character of Z(g),which we also denote by $(λ). In particular, we find that Z(g) acts on theVerma module Mλ−ρ through the central character $(λ).

Proposition 9.4.2 Let M be any g-module. Consider Ind(M) as a moduleover the center Z(g) of Uκc

(g). Then Ind(M) is scheme theoretically supportedon the subscheme OpRS

LG(D) ⊂ OpLG(D×), i.e., it is annihilated by the idealof OpRS

LG(D) in Z(g) ' FunOpLG(D×).Furthermore, if Z(g) acts on M through the character $(λ), then Ind(M)

is scheme theoretically supported on the subscheme OpRSLG(D)$(−λ).

Proof. Define an action of L0 = −t∂t in such a way that it acts by 0 on thegenerating subspace M ⊂ Ind(M), and is compatible with the action of L0 ongκc . Then Ind(M) acquires a Z-grading, which takes only non-negative valueson Ind(M) with the degree 0 part being the subspace M . This implies thatthe subalgebra Z(g)<0 of Z(g) which is spanned by all elements of negativedegrees acts by 0 on Ind(M). The operator L0 also defines a Z-grading onFunOpLG(D×), and the isomorphism Z(g) ' FunOpLG(D×) is compatiblewith this Z-grading. Hence Z(g)<0 is mapped under this isomorphism to thesubalgebra Fun OpLG(D×)<0 spanned by all elements of negative degrees.

But the ideal of OpLG(D×)<0 in FunOpLG(D×) is precisely the ideal ofthe subspace OpRS

LG(D×) ⊂ OpLG(D×) of opers with regular singularities, asshown in Section 9.4.1. This implies the first statement of the proposition.

Next, consider the action of the degree 0 subalgebra Z(g)0 of Z(g) onInd(M). It factors through the quotient

Z ′ = Z(g)0/(Z(g) · Z(g)<0)0.

It follows from Proposition 9.4.1 that Z ′ is isomorphic to the center Z(g)of U(g) and its action on M ⊂ Ind(M), factors through this isomorphism.Furthermore, from Proposition 9.4.1 we have a commutative diagram

Z ′∼−−−−→ FunOpRS

LG(D)y x(Fun h∗)W ∼−−−−→ (Fun h∗)W

where the left vertical arrow is the composition of the isomorphism Z ′ ' Z(g)and the Harish-Chandra homomorphism, the right vertical arrow is inducedby the residue map of Section 9.1.1

res : OpRSLG(D)→ h∗/W,

and the lower horizontal arrow is given by f 7→ f−, f−(λ) = f(−λ). Thisimplies the second statement of the proposition.


In particular, we obtain that the Verma module Mµ,κc is scheme theoreti-cally supported on the subscheme OpRS

LG(D)$(−µ−ρ).

9.4.3 Wakimoto modules and categories of g-modules

Consider the Lie subalgebra

n+ = (n+ ⊗ 1)⊕ (g ⊗ tC[[t]])

of g((t)) and gκc . The corresponding subgroup of G((t)) is denoted by I0. Itis the preimage of the Lie subgroup N+ ⊂ G corresponding to n+ ⊂ g underthe evaluation map G[[t]]→ G. Note that I0 = [I, I], where I is the Iwahorisubgroup defined in Section 10.2.2.

For a central character $(ν) ∈ SpecZ(g) ' h∗/W , let O$(ν) be the blockof the category O of g-modules (with respect to a fixed Borel subalgebra b)whose objects are the g-modules on which Z(g) acts through the character$(ν). Let gκc -modI0

$(−ν) be the category of gκc -modules M such that

• the action of the Lie subalgebra n+ on M is locally nilpotent;• the action of Z(g) on M factors through FunOpRS

LG(D)$(−ν).

On such modules the action of n+ exponentiates to an action of I0, andthis explains the notation.

Then, according to Proposition 9.4.2, the induction functor Ind gives riseto a functor

Ind : O$(ν) → gκc -modI0

$(−ν) .

In particular, for any λ ∈ h∗ the Verma modules Mw(λ+ρ)−ρ,κc, w ∈W , are

objects of the category gκc -modI0

$(−λ−ρ). Moreover, it is easy to see that any

object of gκc -modI0

$(−λ−ρ) has a filtration such that the associated quotientsare quotients of these Verma modules.

Now we want to show that some of the Wakimoto modules of critical levelare also objects of the category gκc

-modI0

$(−λ−ρ). The general construction ofthese modules and the corresponding action of the center Z(g) on them aredescribed in Theorem 8.3.3.

Recall from Section 9.3.1 the subspace Conn(Ωρ)−λD ⊂ Conn(Ωρ)D× . Un-

der the isomorphism Conn(Ωρ)D× ' Conn(Ω−ρ)D× it becomes the subspaceConn(Ω−ρ)λ

D of connections of the form

∂t +λ

t+ u(t), u(t) ∈ Lh[[t]] (9.4.4)

on Ω−ρ. Note that Fun Conn(Ω−ρ)λD is a smooth module over the algebra

FunConn(Ω−ρ)D× . In fact, Fun Conn(Ω−ρ)λD is naturally identified with the


module π0λ over the commutative vertex algebra π0 (see Section 6.2.1). There-

fore we obtain a gκc-module structure on

Wλdef= Mg ⊗ FunConn(Ω−ρ)λ

D. (9.4.5)

We claim that Wλ is an object of the category gκc -modI0

$(−λ−ρ).Indeed, using the explicit formulas for the action of gκc given in Theo-

rem 6.1.6 one checks that the Lie subalgebra n+ acts on Wλ locally nilpo-tently. On the other hand, according to Section 9.3.1, the restriction of theMiura transformation to Conn(Ω−ρ)λ

D = Conn(Ωρ)−λD takes values in

OpRSLG(D)$(−λ−ρ) ⊂ OpLG(D×).

Therefore we obtain from Theorem 8.3.3 that the action of Z(g) on Wλ factorsthrough the quotient FunOpRS

LG(D)$(−λ−ρ). Hence Wλ is an object of the

category gκc -modI0

$(−λ−ρ).

This implies that any quotient of Wλ is also an object of gκc -modI0

$(−λ−ρ).The algebra Fun Conn(Ω−ρ)λ

D acts on Wλ by endomorphisms which commutewith the action of gκc

. Given ∇ ∈ Conn(Ω−ρ)λD, let W∇ be the quotient of Wλ

by the maximal ideal of FunConn(Ω−ρ)λD corresponding to the point ∇. This

is just the Wakimoto module Mg ⊗ C∇ obtained from the one-dimensionalmodule C∇ over FunConn(Ω−ρ)D× corresponding to ∇ ∈ Conn(Ω−ρ)λ

D ⊂Conn(Ω−ρ)D× . According to Theorem 8.3.3, the center Z(g) acts on W∇ viathe central character µ(∇), where µ is the Miura transformation.

For χ ∈ OpRSLG(D), let gκc -modI0

χ be the category of gκc -modules M suchthat

• the action of the Lie subalgebra n+ on M is locally nilpotent;• the center Z(g) acts on M via the character Z(g) → C, corresponding toχ.

Suppose that the residue of χ is $(−λ − ρ) ∈ h∗/W for some λ ∈ h∗.Then the quotients Mw(λ+ρ)−ρ,κc

(χ) of the Verma modules Mw(λ+ρ)−ρ,κc, w ∈

W , by the central character corresponding to χ are objects of this category.Moreover, it is easy to see that any object of gκc -modI0

χ has a filtration suchthat the associated quotients are quotients of Mw(λ+ρ)−ρ,κc

(χ).

The Wakimoto module W∇ is an object of gκc -modI0

χ if and only if ∇ ∈µ−1(χ). According to the isomorphism (9.3.6) (in which we replace G by LG

and Ωρ by Ω−ρ, changing the sign of the residue of the connection), this meansthat

∇ ∈ Conn(Ω−ρ)w(λ+ρ)−ρD , w ∈W. (9.4.6)

The Cartan subalgebra h ⊗ 1 acts diagonally on W∇, but its action may beexponentiated to the action of the corresponding group H (and so W∇ is


I-equivariant) if and only if λ belongs to the set P+ of dominant integralweights.

So let us assume that λ ∈ P+. Then

χ ∈ OpRSLG(D)$(−λ−ρ) = Opnilp,λ

LG

(see Section 9.2.1). We will discuss the corresponding categories gκc -modI0

χ

in Section 10.4. They will appear as the categories associated by the localLanglands correspondence for loop groups to tamely ramified local systemson D×. The following fact will be very important for us in this context (seeSection 10.4.5).

According to Theorem 9.3.7, the points of µ−1(χ) are in bijection with thepoints of the Springer fiber SpResλ(χ) of Resλ(χ). Therefore the Wakimotomodules W∇, where ∇ ∈ µ−1(χ) give us a family of objects of the categorygκc -modI0

χ parameterized by the points of SpResλ(χ).In particular, if Resλ(χ) = 0, and so χ ∈ Opλ

LG, then we have a familyof Wakimoto modules in gκc -modI0

χ parameterized by the points of the flagvariety LG/LB, which is the Springer fiber Sp0.

Thus, we see that the category gκc -modI0

χ is quite complicated if χ ∈OpRS

LG(D)$(−λ−ρ), where λ ∈ P+. This represents one “extreme” exam-

ple of the categories gκc -modI0

χ . On the opposite “extreme” is the category

gκc-modI0

χ , where χ is as above and λ is a generic element of h∗ such that

〈α,w(λ+ ρ)− ρ〉 6∈Z+, α ∈ ∆+, w ∈W.

In this case the structure of this category is rather simple. Applying Proposi-tion 9.3.1 (in which we again replace G by LG and Ωρ by Ω−ρ, changing thesign of the residue of the connection), we obtain that

µ−1(χ) ∩ Conn(Ω−ρ)w(λ+ρ)−ρD

consists of a single point, which we denote by ∇w. By (9.4.6), this impliesthat there are as many points in µ−1(χ) in this case as the number of elementsin the Weyl group W , one in each of the subspaces Conn(Ω−ρ)w(λ+ρ)−ρ

D (beinggeneric, λ is automatically regular). Therefore the category gκc -modI0

χ has|W | inequivalent Wakimoto modules W∇w

, w ∈W , in this case.For each w ∈W we have a non-zero homomorphism Mw(λ+ρ)−ρ,κc

→W∇w.

In the special case when these modules are Z-graded (which means that χ isinvariant under the vector field L0 = −t∂t), we have shown in Section 6.4that all of them are irreducible and this homomorphism is an isomorphism.Therefore it follows the same is true for generic χ (we expect that this istrue for all χ with the residue $(−λ − ρ) with λ as above). We find that inthis case the category gκc -modI0

χ has |W | non-isomorphic irreducible objectsW∇w

, w ∈W . Analyzing the action of the Cartan subalgebra h⊗ 1 on them,

9.5 Endomorphisms of the Verma modules 273

we find that there can be no non-trivial extensions between them. Thus, anyobject of the category gκc

-modI0

χ is a direct sum of copies of the irreducibleWakimoto modules W∇w

, w ∈W .

For a general λ ∈ h∗ the structure of the category gκc -modI0

χ with χ ∈OpRS

LG(D)$(−λ−ρ) is intermediate between the two extreme cases analyzedabove.

9.5 Endomorphisms of the Verma modules

In this section and the next we identify the algebras of endomorphisms of someinduced g-modules of critical level with algebras of functions on subvarietiesof OpLG(D×) constructed earlier in this chapter.

Consider first the case of the vacuum module Vκc(g). According to formula

(3.3.3), we have

EndbgκcVκc(g) ' z(g).

Therefore, by Theorem 4.3.2, we have the isomorphism

EndbgκcVκc(g) ' FunOpLG(D). (9.5.1)

Moreover, we find that the homomorphism Z(g)→ EndbgκcVκc(g) is surjective

and corresponds to the homomorphism

FunOpLG(D×)→ FunOpLG(D).

Thus, Vκc(g) corresponds to the subvariety

OpLG(D) ⊂ OpLG(D×).

Here we generalize this result to the case of Verma modules, which turn outto correspond to the subvarieties OpRS

LG(D)$, and in the next section to thecase of Weyl modules, which correspond to Opλ

LG.

9.5.1 Families of Wakimoto modules

We consider first the Verma modules. Recall that a Verma module of levelκ and highest weight λ ∈ h∗ is defined as the induced module (6.3.9). Sincewe will consider exclusively the critical level κ = κc, we will drop the subscriptκc from the notation and write simply Mλ.

Note that Mλ = Ind(Mλ), where Ind is the induction functor introduced inSection 9.4.2.

We wish to describe the algebra of endomorphisms of Mλ for all highestweights λ ∈ h∗. In order to do this, we will identify each Verma module witha particular Wakimoto module, for which the algebra of endomorphisms iseasy to compute. An example of such an identification is given by Propo-sition 6.3.3. This proposition suggests that we need to use more general


Wakimoto modules than the modules Wλ = Mg ⊗ πλ considered above (see,e.g., Section 9.4.3). More precisely, we will use the Wakimoto obtained by thesemi-infinite parabolic induction of Section 6.3 with respect to a Borel sub-algebra of g of the form wb−w

−1, where b− is the standard Borel subalgebrathat we have used before and w is an element of the Weyl group of g.

In this section we define a family of Wakimoto modules for each element wof the Weyl group of g. This generalizes the construction of the family Wλ,which corresponds to the case w = 1 (see Section 9.4.3), along the lines of thegeneral construction presented in Section 6.3.

Recall the Weyl algebra Ag introduced in Section 5.3.3. We define the Ag-module Mw

g as the module generated by the vacuum vector |0〉w such that

aα,n|0〉w = a∗α,n|0〉w = 0, n > 0,

aα,0|0〉w = 0, α ∈ ∆+ ∩ w−1(∆+); a∗α,0|0〉w = 0, α ∈ ∆+ ∩ w−1(∆−).

According to Theorem 8.3.3, for each λ ∈ h∗ the tensor product Mwg ⊗

FunConn(Ω−ρ)λD is a gκc -module. Denote by ω the corresponding homo-

morphism

gκc → End(Mwg ⊗ FunConn(Ω−ρ)λ

D).

We define a new gκc -module, denoted by Wwλ , as the w−1-twist of Mw

g ⊗FunConn(Ω−ρ)λ

D. More precisely, we define a new action of gκc by the formulax 7→ ω(Ad w−1 · x), where w is the Tits lifting of w to the group G.

In particular, W 1λ is the module Wλ introduced in Section 9.4.3.

Recall that given a gκc -module on which the operator L0 = −t∂t and theLie subalgebra h ⊗ 1 ⊂ gκ act diagonally with finite-dimensional commoneigenspaces, we define the character of M as the formal series

chM =∑

bλ∈(h⊕CL0)∗

dimM(λ) ebλ, (9.5.2)

where M(λ) is the eigenspace of L0 and h ⊗ 1 corresponding to the weightλ : (h⊕ CL0)→ C.

Let us compute the character of the module Wwλ . The operator L0 on

Mwg ⊗ FunConn(Ω−ρ)λ

D is the sum of the corresponding operators on Mwg

and Fun Conn(Ω−ρ)λD. The former is defined by the commutation relations

[L0, aα,n] = −naα,n, [L0, a∗α,n] = −na∗α,n,

and the condition that L0|0〉w = 0. The latter is determined from the actionof the vector field −t∂t on the space Conn(Ω−ρ)λ

D. If we write an element ofConn(Ω−ρ)λ

D in the form (9.4.4), and expand ui(t) = 〈αi,u(t)〉 in the series

ui(t) =∑n<0

ui,nt−n−1,


we obtain an identification

FunConn(Ω−ρ)λD ' C[ui,n]i=1,...,`;n<0.

Then L0 is just the derivation of this polynomial algebra completely deter-mined by the formulas

L0 · ui,n = −nui,n.

Next, we consider the action of h = h⊗ 1 ⊂ gκc on Mwg . It follows from the

original formulas given in Theorem 6.1.6 that for any h ∈ h we have

[h, aα,n] = 〈α, h〉aα,n, [h, a∗α,n] = −〈α, h〉a∗α,n.

After we twist the action of gκc , and hence of h, by w we obtain new relations

[h, aα,n] = 〈w(α), h〉aα,n, [h, a∗α,n] = −〈w(α), h〉a∗α,n. (9.5.3)

Note that for any µ ∈ h∗ we have 〈µ,w−1(h)〉 = 〈w(µ), h〉.To compute the action of h on |0〉w ⊗ 1 ∈ Mw

g ⊗ 1, observe that accordingto the formula for the action of hi,0 given in Theorem 6.1.6 we have

hi,0 · |0〉w ⊗ 1 =

〈w(λ), hi〉 −∑

β∈∆+

〈w(β), hi〉a∗β,0aβ,0

|0〉w ⊗ 1

=

〈w(λ), hi〉 −∑

β∈∆+∩w−1(∆−)

〈w(β), hi〉

|0〉w ⊗ 1

= 〈w(λ)− w(w−1(ρ)− ρ), hi〉|0〉w ⊗ 1

= 〈w(λ+ ρ)− ρ, hi〉|0〉w ⊗ 1.

Therefore the h-weight of the vector |0〉w ⊗ 1 is equal to w(λ+ ρ)− ρ.Putting all of this together, we obtain the following formula for the character

of the module Wwλ :

chWwλ = ew(λ+ρ)−ρ

∏α∈b∆+

(1− e−α)−mult(α), (9.5.4)

where ∆+ is the set of positive roots of g. Here the terms correspondingto the real roots come from the generators aα,n and a∗α,n of Ag, and theterms corresponding to the imaginary roots come from the generators ui,n

of FunConn(Ω−ρ)λD. Thus we obtain that the character of Ww

λ is equal tothe character of the Verma module Mw(λ+ρ)−ρ with the highest weight w(λ+ρ) − ρ, whose character has been computed in the course of the proof ofProposition 6.3.3.


9.5.2 Verma modules and Wakimoto modules

Observe now that any weight ν of g may be written (possibly, in different ways)in the form ν = w(λ+ρ)−ρ, where λ has the property that 〈λ, α〉 6∈Z+ for anyα ∈ ∆+. Note that w is uniquely determined if ν+ρ is in the Weyl group orbitof a regular dominant integral weight. The following assertion establishes anisomorphism between the Verma module Mw(λ+ρ)−ρ and a Wakimoto moduleof a particular type.

Proposition 9.5.1 For any weight λ such that 〈λ, α〉 6∈Z+ for all α ∈ ∆+,and any element w of the Weyl group of g, the Verma module Mw(λ+ρ)−ρ isisomorphic to the Wakimoto module Ww

w(λ+ρ)−ρ.

Proof. In the case when w = w0 and λ = −2ρ this is proved in Propo-sition 6.3.3 (note that W+

0,κc= Ww0

−2ρ). We will use a similar argument ingeneral.

First, observe that the character formula implies that the vector |0〉w ⊗ 1 ∈Ww

w(λ+ρ)−ρ is a highest weight vector of highest weight w(λ+ρ)−ρ. Thereforethere is a canonical homomorphism

Mw(λ+ρ)−ρ →Www(λ+ρ)−ρ

taking the highest weight vector of Mw(λ+ρ)−ρ to |0〉w⊗1 ∈Www(λ+ρ)−ρ. Since

the characters of Mw(λ+ρ)−ρ and Www(λ+ρ)−ρ are equal, the proposition will

follow if we show that this homomorphism is surjective, or, equivalently, thatWw

w(λ+ρ)−ρ is generated by its highest weight vector |0〉w ⊗ 1.Suppose that Ww

w(λ+ρ)−ρ is not generated by the highest weight vector.Then there exists a homogeneous linear functional on Ww

w(λ+ρ)−ρ, whoseweight is less than the highest weight (λ, 0) ∈ h∗ ⊕ (CL0)∗ and which isinvariant under the Lie subalgebra

n− = n− ⊗ C[t−1]⊕

b+ ⊗ C[t−1],

and in particular, under its Lie subalgebra

L−nw+ = (wn+w

−1) ∩ n− ⊗ C[t−1]⊕

(wn+w−1) ∩ n+ ⊗ t−1C[t−1].

Therefore this functional factors through the space of L−nw+-coinvariants of

Www(λ+ρ)−ρ. But it follows from the formulas for the action of gκc

onWww(λ+ρ)−ρ

that the action of the Lie subalgebra L−nw+ is free, and the space of coinvari-

ants is isomorphic to

C[a∗α,n]α∈∆+,n≤0 ⊗ FunConn(Ω−ρ)w0(λ+ρ)−ρD .

Our functional has to take a non-zero value on a weight vector in thissubspace of weight other than the highest weight w(λ+ρ)−ρ. Using formula(9.5.3), we obtain that the weight of a∗α,n is n− w(α). Therefore the weights


of vectors in this subspace are of the form

(w(λ+ ρ)− ρ) +Nδ −∑

j

γj , (9.5.5)

where N ≤ 0 and each γj is a root in w(∆+). If such a functional exists, thenWw

w(λ+ρ)−ρ has an irreducible subquotient of such highest weight. Since thecharacters of irreducible highest weight modules are linearly independent, andthe characters of Ww

w(λ+ρ)−ρ and Mw(λ+ρ)−ρ are equal, we obtain that thenMw(λ+ρ)−ρ must also have an irreducible subquotient whose highest weight isof the form (9.5.5).

Now recall the Kac–Kazhdan theorem [KK] describing the set of highestweights of irreducible subquotients of Verma modules (see Proposition 6.3.3).A weight µ = (µ, n) appears in the decomposition of Mbν , where ν = (ν, 0) ifand only n ≤ 0 and either µ = 0 or there exists a finite sequence of weightsµ0, . . . , µm ∈ h∗ such that µ0 = µ, µm = ν, µi+1 = µi±miβi for some positiveroots βi and positive integers mi which satisfy

〈µi + ρ, βi〉 = mi. (9.5.6)

Let us denote the set of such weights µ = (µ, n) with a fixed weight ν = (ν, 0)by S(ν). It follows from the above description that if (µ, n) ∈ S(ν), thenn ≤ 0, µ belongs the W -orbit of ν with respect to the ρ-shifted action, andmoreover for any µ ∈ S(ν), the difference µ − ν is a linear combination ofroots with integer coefficients.

Suppose now that a weight λ satisfies the property in the statement ofthe proposition that 〈λ, α〉 6∈Z+ for any α ∈ ∆+. This implies that if µ =y(λ+ ρ)− ρ and µ− λ =

∑i∈I niαi, where ni ∈ Z, then we necessarily have

ni ≥ 0. Therefore for any weight (µ, 0) ∈ S(λ) we have µ = λ+∑

j βj , whereβj ∈ ∆+. This implies that if µ′ = (µ′, n) ∈ S(w(λ+ ρ)− ρ), then we have

µ′ = (w(λ+ ρ)− ρ)−Nδ +∑

j

βj , βj ∈ w(∆+), N ≤ 0.

Such a weight cannot be of the form (9.5.5), unless it is w(λ+ρ)−ρ). ThereforeWw

w(λ+ρ)−ρ is generated by its highest weight vector and hence Www(λ+ρ)−ρ is

isomorphic to Mw(λ+ρ)−ρ.

9.5.3 Description of the endomorphisms of Verma modules

We now use Proposition 9.5.1 to describe the algebra EndbgκcMν of endomor-

phisms of the Verma module Mν . Observe that since Mν is freely generatedfrom its highest weight vector, such an endomorphism is uniquely determinedby the image of the highest weight vector vector. The image of the highestweight vector can be an arbitrary n+-invariant vector which has weight νwith respect to the Cartan subalgebra. We denote the space of such vectors


by (Mν)bn+ν . Using the same argument as in the proof of the isomorphism

(3.3.3), we obtain that

EndbgκcMν ' (Mν)bn+

ν . (9.5.7)

Thus, for any λ that satisfies the conditions of Proposition 9.5.1 we obtainthe following isomorphisms of vector spaces:

EndbgκcMw(λ+ρ)−ρ ' (Mw(λ+ρ)−ρ)

bn+

w(λ+ρ)−ρ

' (Www(λ+ρ)−ρ)

bn+

w(λ+ρ)−ρ. (9.5.8)

Lemma 9.5.2 For any λ ∈ h∗ the space of n+-invariant vectors of weightw(λ+ ρ)− ρ in Ww

w(λ+ρ)−ρ is equal to its subspace |0〉w ⊗ FunConn(Ω−ρ)λD.

Proof. It follows from the formulas of Theorem 6.1.6 for the actionof gκc on the Wakimoto modules that all vectors in the subspace |0〉w ⊗FunConn(Ω−ρ)w(λ+ρ)−ρ

D are annihilated by n+ and have weight w(λ+ ρ)− ρ.Let us show that there are no other n+-invariant vectors in Ww

w(λ+ρ)−ρ of thisweight.

An n+-invariant vector is in particular annihilated by the Lie subalgebra

L+nw+ = (wn+w

−1 ∩ n−)⊗ tC[t]⊕

(wn+w−1 ∩ n+)⊗ C[t].

But it follows from the formulas of Theorem 6.1.6 that this Lie algebra actscofreely on Mw

g ⊗FunConn(Ω−ρ)λD and so the character of the space of L+nw

+-invariants is equal to the ratio between the character of Ww

w(λ+ρ)−ρ and thecharacter of the restricted dual of the universal enveloping algebra U(L+nw

+).This ratio is equal to

ew(λ+ρ)−ρ∏n>0

(1− e−nδ)−`∏

α∈w(∆−)∩∆−,n≥0

(1− e−nδ−α)−1×

×∏

α∈w(∆−)∩∆+,n>0

(1− e−nδ+α)−1.

The character of the weight w(λ + ρ) − ρ subspace of the space of L+nw+-

invariants in Mwg ⊗ FunConn(Ω−ρ)λ

D is therefore equal to

ew(λ+ρ)−ρ∏n>0

(1− e−nδ)−`,

which is precisely the character of the subspace |0〉w⊗FunConn(Ω−ρ)λD. This

completes the proof.

Recall from Theorem 8.3.3 that the action of the center Z(g) on Www(λ+ρ)−ρ

factors through the homomorphism

Z(g) ' FunOpLG(D×)→ FunConn(Ω−ρ)D×

9.6 Endomorphisms of the Weyl modules 279

induced by the Miura transformation. Therefore it factors through the homo-morphism

FunOpLG(D×)→ FunConn(Ω−ρ)λD.

But we know from the discussion of Section 9.3.1 that this homomorphismfactors through the homomorphism

FunOpRSLG(D)$(−λ−ρ) → FunConn(Ω−ρ)λ

D.

Suppose that λ satisfies the condition of Proposition 9.5.1. Then this map isan isomorphism by Proposition 9.3.1.† Combining this with the isomorphism(9.5.8) and the statement of Lemma 9.5.2, we obtain the following result (see[FG2], Corollary 13.3.2).

Theorem 9.5.3 The center Z(g) maps surjectively onto EndbgκcMν , and we

have a commutative diagram

Z(g) ∼−−−−→ FunOpLG(D×)y yEndbgκc

Mν∼−−−−→ FunOpRS

LG(D)$(−ν−ρ)

for any weight ν.Furthermore, the Verma module Mν is free over the algebra

EndbgκcMν ' FunOpRS

LG(D)$(−ν−ρ).

9.6 Endomorphisms of the Weyl modules

Let Vλ be a finite-dimensional representation of the Lie algebra g with highestweight λ, which is a dominant integral weight of g. We extend the action ofg to an action of g⊗C[[t]]⊕ 1 in such a way that g⊗ tC[[t]] acts trivially and1 acts as the identity. Let Vλ,κ be the induced gκ-module

Vλ,κ = U(gκ) ⊗U(g[[t]]⊕C1)

Vλ.

This is the Weyl module of level κ with highest weight λ.In what follows we will only consider the Weyl modules of critical level

κ = κc, so we will drop the subscript κc and denote them simply by Vλ. Inparticular, V0 = Vκc

(g), the vacuum module.We note that Vλ = Ind(Vλ), where Ind is the induction functor introduced

in Section 9.4.2. Since Vλ is the finite-dimensional quotient of the Vermamodule Mλ over g, and we have Ind(Mλ) = Mλ, we obtain a homomorphism

Mλ,κc → Vλ.

† Note that in order to apply the statement of Proposition 9.3.1 we need to replace G byLG and Ωρ by Ω−ρ, which changes the sign of the residue of the connection.


It is clear from the definitions that this homomorphism is surjective, and sothe Weyl module Vλ is realized as a quotient of the Verma module Mλ.

9.6.1 Statement of the result

Recall the space OpλLG of LG-opers with regular singularity, residue $(−ρ−λ)

and trivial monodromy, introduced in Section 9.2.3. In this section we willprove the following result, which generalizes the isomorphism (9.5.1) corre-sponding to λ = 0. For g = sl2 it was proved in [F3], and for an arbitraryg in [FG6] (the proof presented below is similar to that proof). It was alsoindependently conjectured by Beilinson and Drinfeld (unpublished).

Theorem 9.6.1 For any dominant integral weight λ the center Z(g) mapssurjectively onto Endbgκc

Vλ, and we have the following commutative diagram

Z(g) ∼−−−−→ FunOpLG(D×)y yEndbgκc

Vλ∼−−−−→ FunOpλ

LG

Our strategy of the proof will be as follows: we will first construct a homo-morphism Vλ →Wλ. Thus, we obtain the following maps:

Mλ Vλ →Wλ, (9.6.1)

and the corresponding maps of the algebras of endomorphisms

EndbgκcMλ → Endbgκc

Vλ → EndbgκcWλ. (9.6.2)

According to Theorem 9.5.3 and the argument used in Lemma 6.3.4, thecompsite map is the homomorphism of the algebras of functions correspondingto the morphism

µλ : Conn(Ω−ρ)λD → OpRS

LG(D)$(−λ−ρ),

obtained by restriction from the Miura transformation. We therefore find thatthe image of this composite map is precisely Fun Opλ

LG ⊂ FunConn(Ω−ρ)λD =

EndbgκcWλ.

On the other hand, we show that the second map in (9.6.2) is injective andits image is contained in FunOpλ

LG. This implies that EndbgκcVλ is isomorphic

to FunOpλLG.

9.6.2 Weyl modules and Wakimoto modules

Let us now proceed with the proof and construct a homomorphism Vλ →Wλ.Recall that the Wakimoto module Wλ was defined by formula (9.4.5):

Wλ = Mg ⊗ FunConn(Ω−ρ)λD. (9.6.3)


In order to construct a map Vλ → Wλ, we observe (as we did previously, inSections 6.2.3 and 9.4.1) that the action of the constant subalgebra g ⊂ gκ onthe subspace

W 0λ = C[a∗α,0]α∈∆+ |0〉 ⊗ 1 ⊂Mg ⊗ FunConn(Ω−ρ)λ

D

coincides with the natural action of g on the contragredient Verma moduleM∗λ realized as FunN+ = C[yα]α∈∆+ , where we substitute a∗α,0 7→ yα (seeSection 5.2.4) for the definition of this action). In addition, the Lie subalgebrag⊗t⊗C[[t]] ⊂ gκc acts by zero on the subspace W 0

λ , and 1 acts as the identity.Therefore the injective g-homomorphism Vλ →M∗λ gives rise to a non-zero

gκc-homomorphism

ıλ : Vλ →Wλ,

sending the generating subspace Vλ ⊂ Vλ to the image of Vλ in W 0λ 'M∗λ .

Now we obtain a sequence of maps

Mλ Vλ →Wλ. (9.6.4)

According to formula (9.5.7), Theorem 9.5.3, and Proposition 9.2.1, we have

EndbgκcMλ = (Mλ)bn+

λ ' FunOpRSLG(D)$(−λ−ρ) ' FunOpnilp,λ

LG.

Next, according to Lemma 9.5.2, we have

EndbgκcWλ = (Wλ)bn+

λ = FunConn(Ω−ρ)λD.

Note that by formula (9.6.3), FunConn(Ω−ρ)λD = |0〉 ⊗ FunConn(Ω−ρ)λ

D isnaturally a subspace of Wλ.

Finally, note that

EndbgκcVλ = (Vλ ⊗ V ∗λ )g[[t]] = (Vλ)bn+

λ .

Indeed, any endomorphism of Vλ is uniquely determined by the image ofthe generating subspace Vλ. This subspace therefore defines a g[[t]]-invariantelement of (Vλ ⊗ V ∗λ )g[[t]] and its highest weight vector gives rise to a vectorin (Vλ)bn+

λ . In the same way as before, we obtain that the resulting maps areisomorphisms.

We obtain the following commutative diagram:

(Mλ)bn+λ −−−−→ (Vλ)bn+

λ −−−−→ (Wλ)bn+λy∼ y∼ y∼

EndbgκcMλ −−−−→ Endbgκc

Vλ −−−−→ EndbgκcWλy∼ y∼ y∼

FunOpnilp,λLG

−−−−→ ? −−−−→ FunConn(Ω−ρ)λD

(9.6.5)


This implies that we have a sequence of homomorphisms

EndbgκcMλ → Endbgκc

Vλ → EndbgκcWλ. (9.6.6)

Moreover, it follows that EndbgκcVλ has a subquotient isomorphic to the image

of the homomorphism

FunOpnilp,λLG

→ FunConn(Ω−ρ)λD (9.6.7)

obtained from the above diagram.By construction, the last homomorphism fits into a commutative diagram

Z(g) −−−−→∼

EndbgκcMλ −−−−→ Endbgκc

Wλy y∼ y∼FunOpLG(D×) −−−−→ FunOpnilp,λ

LG−−−−→ FunConn(Ω−ρ)λ

D

and the composition

FunOpLG(D×)→ FunConn(Ω−ρ)λD

of the maps on the bottom row corresponds to the natural action of Z(g)on Wλ. According to Theorem 8.3.3, this map is the Miura transformationmap. Therefore the homomorphism (9.6.7) coincides with the homomorphismobtained by restriction of the Miura transformation. The fact that this ho-momorphism factors through FunOpnilp,λ

LGsimply means that the image of

Conn(Ω−ρ)λD ⊂ Conn(Ω−ρ)D× under the Miura transformation belongs to

Opnilp,λLG

. We already know that from formula (9.3.4) and the diagram (9.3.5).Moreover, we obtain from the diagram (9.6.5) that the image of the homo-morphism

EndbgκcVλ → Endbgκc

Wλ

is nothing but the algebra of functions on the image of Conn(Ω−ρ)λD in Opnilp,λ

LG

under the Miura map. Let us find what this image is.

9.6.3 Nilpotent opers and Miura transformation

Recall the space MOpλ,wLG⊂ MOpλ

LG defined in formula (9.3.11). Considerthe case when w = 1. The corresponding space MOpλ,1

LGparametrizes pairs

(χ,F ′LB,0), where χ = (F ,∇,FLB) ∈ Opnilp,λLG

and F ′LB,0 is a LB-reduction ofthe fiber at 0 ∈ D of the LG-bundle F underlying χ, which is stable underthe residue Resλ(χ) and is in generic relative position with the oper reductionFLB,0. But since Resλ(χ) ∈ LnFLG

, we find that such a reduction F ′LB,0

exists only if Resλ(χ) = 0, so that χ ∈ OpλLG. The space of such reductions

for an oper χ satisfying this property is a torsor over the group LNFLB ,0 (itis isomorphic to the big cell of the flag variety LG/LB).

Thus, we obtain that the image of the forgetful morphism MOpλ,1LG→


Opnilp,λLG

is OpλLG ⊂ Opnilp,λ

LG, and MOpλ,1

LGis a principal LNFLB,0

-bundle overOpλ

LG.On the other hand, according to Theorem 9.3.6, we have an isomorphism

Conn(Ω−ρ)λD ' MOpλ,1

LG.

and the Miura transformation Conn(Ω−ρ)λD → Opnilp,λ

LGcoincides with the

forgetful map MOpλ,1LG→ Opnilp,λ

LG. Thus, we obtain the following:

Lemma 9.6.2 The image of Conn(Ω−ρ)λD in Opnilp,λ

LGunder the Miura trans-

formation is equal to OpλLG. The map Conn(Ω−ρ)λ

D → OpλLG is a principal

LNFLB,0-bundle over Opλ

LG.

Let (Vλ)bn+,0λ be the subspace of (Vλ)bn+

λ = EndbgκcVλ which is the image of

the top left horizontal homomorphism in (9.6.6). Then we obtain the followingcommutative diagram:

(Vλ)bn+λ −−−−→ FunConn(Ω−ρ)λ

Dx x(Vλ)bn+,0

λ −−−−→ FunOpλLG

(9.6.8)

where the vertical maps are injective and the bottom horizontal map is sur-jective. We now wish to prove that the top horizontal map is injective and itsimage is contained in the image of the right vertical map. This would implythat the top horizontal map and the left vertical map are isomorphisms, andhence we will obtain the statement of Theorem 9.6.1.

As the first step, we will now obtain an explicit realization of

FunOpλLG ⊂ FunConn(Ω−ρ)λ

D

as the intersection of kernels of some screening operators, generalizing thedescription in the case λ = 0 obtained in Section 8.2.3.

According to Lemma 9.6.2, FunOpλLG is realized as the subalgebra of the

algebra Fun Conn(Ω−ρ)λD which consists of the LNFLB,0

-invariant, or equiva-lently, LnLB,0-invariant functions. An element of Conn(Ω−ρ)λ

D is representedby an operator of the form

∂t +λ

t+ u(t), u(t) =

∑n≥0

unt−n−1 ∈ Lh[[t]]. (9.6.9)

Therefore

FunConn(Ω−ρ)λD = C[ui,m]i=1,...,`;m<0, (9.6.10)

where ui,m = 〈αi,um〉. Here and below we are using a canonical identificationLh = h∗. We now compute explicitly the action of generators of LnLB,0 on this


space, by generalizing our computation in Section 8.2.3 (which correspondsto the case λ = 0).

For that we fix a Cartan subalgebra Lh in Lb and a trivialization of FLB,0

such that LnFLB,0is identified with Ln, and LbF ′LB,0

is identified with Lb−.

Next, choose generators ei, i = 1, . . . , `, of Ln with respect to the action of Lh

on Ln in such a way that together with the previously chosen fi they satisfy thestandard relations of Lg. The LnFLB,0

-action on Conn(Ω−ρ)λD then gives rise

to an infinitesimal action of the ei’s on Conn(Ω−ρ)λD. We will now compute

the corresponding derivation on FunConn(Ω−ρ)λD.

With respect to our trivialization of FLB,0 the oper connection obtained byapplying the Miura transformation to (9.6.9) reads as follows:†

∂t +∑i=1

tλifi − u(t), u(t) =∑n≥0

unt−n−1 ∈ Lh[[t]], (9.6.11)

where λi = 〈αi, λ〉.The infinitesimal gauge action of ei on this connection operator is given by

the formula

δu(t) = −[xi(t) · ei, ∂t +∑i=1

tλifi − u(t)], (9.6.12)

where xi(t) ∈ C[[t]] is such that xi(0) = 1, and the right hand side of formula(9.6.12) belongs to Lh[[t]]. These conditions determine xi(t) uniquely. Indeed,the right hand side of (9.6.12) reads

−tλixi(t) · αi − ui(t)xi(t) · ei + ∂txi(t) · ei,

where ui(t) = 〈αi,u(t)〉. Therefore it belongs to Lh[[t]] if and only if

∂txi(t) = ui(t)xi(t). (9.6.13)

If we write

xi(t) =∑n≤0

xi,nt−n,

and substitute it into formula (9.6.13), we obtain that the coefficients xi,n

satisfy the following recurrence relation:

nxi,n = −∑

k+m=n;k<0;m≤0

ui,kxi,m, n < 0.

We find from this formula and the condition xi(0) = 1 that∑n≤0

xi,nt−n = exp

(∑m>0

ui,−m

mtm

). (9.6.14)

† The minus sign in front of u(t) is due to the fact that before applying the Miura trans-formation we switch from connections on Ω−ρ to connections on Ωρ.


Now we obtain that

δu(t) = −tλixi(t) · αi,

and so

δuj(t) = −tλiajixi(t),

where aji = 〈αj , αi〉 is the (j, i) entry of the Cartan matrix of g. In otherwords, the operator ei acts on the algebra Fun Conn(Ω−ρ)λ

D = C[ui,n] by thederivation

Vi[λi + 1] = −∑j=1

aji

∑n≥λi

xi,n−λi

∂

∂uj,−n−1, (9.6.15)

where the xi,n’s are given by formula (9.6.14).Note that when λi = 0, i = 1, . . . , `, this operator, which appeared in

formula (8.2.10) (up to the substitution ui,n 7→ −ui,n, aji 7→ aij , becausethere we were considering G-opers and connections on Ωρ), was identifiedwith a limit of the (−1)st Fourier coefficient of a vertex operator betweenFock representations of the Heisenberg vertex algebra (see Section 8.1.2). Wewill show in the proof of Proposition 9.6.7 that the operator (9.6.15) maybe identified with the limit of the −(λi + 1)st Fourier coefficient of the samevertex operator.

Now recall that we have identified FunOpλLG with the algebra of Ln-invariant

functions on Conn(Ω−ρ)λD. These are precisely the functions that are anni-

hilated by the generators ei, i = 1, . . . , `, of Ln, which are given by formula(9.6.15). Therefore we obtain the following characterization of FunOpλ

LG asa subalgebra of FunConn(Ω−ρ)λ

D.

Proposition 9.6.3 The image of FunOpλLG in FunConn(Ω−ρ)λ

D under theMiura map is equal to the intersection of the kernels of the operatorsVi[λi + 1], i = 1, . . . , `, given by formula (9.6.15).

As a corollary of this result, we obtain a character formula for FunOpλLG

with respect to the Z+-grading defined by the operator L0 = −t∂t. Accordingto Lemma 9.6.2, the action of LN generated by the operators (9.6.15) onConn(Ω−ρ)λ

D is free. Therefore we have an isomorphism of vector spaces

FunConn(Ω−ρ)λD ' FunOpλ

LG⊗Fun LN.

According to formula (9.6.15), the operator ei is homogeneous of degree

−(λi + 1) = −〈αi, λ+ ρ〉

with respect to the Z+-grading introduced above. Therefore the degree of theroot generator eα, α ∈ ∆+, of Ln is equal to −〈α, λ+ ρ〉. Hence we find that


the character of Fun LN is equal to∏α∈∆+

(1− q〈α,λ+ρ〉)−1.

On the other hand, since deg ui,n = −n, we find from formula (9.6.10) thatthe character of FunConn(Ω−ρ)λ

D is equal to∏n>0

(1− qn)−`.

Therefore we find that the character of FunOpλLG is the ratio of the two, which

we rewrite in the form∏α∈∆+

1− q〈α,λ+ρ〉

1− q〈α,ρ〉

∏i=1

∏ni≥di+1

(1− qni)−1, (9.6.16)

using the identity

∏α∈∆+

(1− q〈α,ρ〉) =∏i=1

di∏mi=1

(1− qmi).

9.6.4 Computations with jet schemes

In order to complete the proof of Theorem 9.6.1 it would suffice to show thatthe character of

EndbgκcVλ = (Vλ ⊗ V ∗λ )g[[t]]

is less than or equal to the character of FunOpλLG given by formula (9.6.16).

In the case when λ = 0 we have shown in Theorem 8.1.3 that the natural mapgr(V0)g[[t]] → (gr V0)g[[t]] is an isomorphism. This makes it plausible that themap

gr(Vλ ⊗ V ∗λ )g[[t]] → (gr(Vλ ⊗ V ∗λ ))g[[t]] (9.6.17)

is an isomorphism for all λ. If we could show that the character of (gr(Vλ ⊗V ∗λ ))g[[t]] is less than or equal to the character of FunOpλ

LG, then this wouldimply that the map (9.6.17) is an isomorphism, as well as prove Theorem 9.6.1,so we would be done.

In this section and the next we will compute the character of (gr(Vλ ⊗V ∗λ ))g[[t]]. However, we will find that for general λ this character is greaterthan the character of FunOpλ

LG. The equality is achieved if and only if λ is aminuscule weight. This means that the map (9.6.17) is an isomorphism if andonly if λ is minuscule. Therefore this way we are able to prove Theorem 9.6.1for minuscule λ only. Nevertheless, the results presented below may be ofindependent interest. In order to prove Theorem 9.6.1 for a general λ we willuse another argument given in Section 9.6.6.


Thus, our task at hand is to compute the character of (gr(Vλ ⊗ V ∗λ ))g[[t]].The PBW filtration on Uκc

(g) gives rise to a filtration on Vλ considered as thegκc

-module generated from the subspace Vλ. The corresponding associatedgraded space is

gr Vλ = Vλ ⊗ Sym(g((t))/g[[t]]) ' Vλ ⊗ Fun g∗[[t]].

Therefore we obtain an embedding

(gr(Vλ ⊗ V ∗λ ))g[[t]] → (Vλ ⊗ V ∗λ ⊗ Fun g∗[[t]])g[[t]].

Thus, we can find an upper bound on the character of EndbgκcVλ by computing

the character of (Vλ ⊗ V ∗λ ⊗ Fun g∗[[t]])g[[t]].The first step is the computation of the algebra of invariants in Fun g∗[[t]]

under the action of the Lie algebra tg[[t]] ⊂ g[[t]].Recall the space

P = g∗/G = Spec(Fun g∗)G ' Spec C[P i]i=1,...,`,

and the morphism

p : g∗ → P.

Consider the corresponding jet schemes

JNP = Spec C[P i,n]i=1,...,`;−N−1≤n<0,

introduced in the proof of Theorem 3.4.2. Here and below we will allow N tobe finite or infinite (the definition of jet schemes is given in Section 3.4.2.

According to Theorem 3.4.2, we have

(FunJNg∗)JN G = C[P i,n]i=1,...,`;−N−1≤n<0.

In other words, the morphism

JNp : JNg∗ → JNP

factors through an isomorphism

JNg∗/JNG ' JNP,

where, by definition,

JNg∗/JNG = Spec(FunJNg∗)JN G.

We have natural maps g∗ → P and JNP → P. Denote by JNP the fiberproduct

JNP = JNP ×P

g∗.

By definition, the algebra Fun JNP of functions on JNP is generated byFunJNP = (FunJNg∗)JN G and Fun g∗, with the only relation being that


we identify elements of (Fun JNg∗)JN G and Fun g∗ obtained by pull-back ofthe same element of (Fun g∗)G.

Consider the morphism

JNp : JNg∗ → JNP,A(t) 7→ (JNp(A(t)), A(0))

and the corresponding homomorphism of algebras

Fun JNP → FunJNg∗. (9.6.18)

Let JNG(1) be the subgroup of JNG of elements congruent to the identity

modulo t. Note that the Lie algebra of JNG is JNg = g⊗ (C[[t]]/tN+1C[[t]])and the Lie algebra of JNG

(1) is JNg(1) = g ⊗ (tC[[t]]/tN+1C[[t]]). For allalgebras that we consider in the course of this discussion the spaces of in-variants under JNG (resp., JNG

(1)) are equal to the corresponding spaces ofinvariants under JNg (resp., JNg(1)).

It is clear that the image of the homomorphism (9.6.18) consists of elementsof FunJNg∗ that are invariant under the action of JNg(1) (and hence JNG

(1)).

Proposition 9.6.4 The homomorphism (9.6.18) is injective and its imageis equal to the algebra of JNg(1)-invariants in FunJNg∗ (for both finite andinfinite N).

Proof. As in the proof of Theorem 3.4.2, consider the subset g∗reg ofregular elements in g∗. Then we have a smooth and surjective morphismpreg : g∗ → P. The corresponding morphism JNpreg : JNg∗reg → JNP is alsosmooth and surjective. It follows from the definition of jet schemes that

JNg∗reg ' g∗reg ×(g∗ ⊗ (tC[[t]]/tN+1C[[t]])

).

Therefore JNg∗reg is a smooth subscheme of JNg∗, which is open and dense,and its complement has codimension greater than 1.

As explained in the proof of Theorem 3.4.2, each fiber of JNpreg consists ofa single orbit of the group JNG. We used this fact to derive that

(FunJNg∗)JN G = (Fun JNg∗reg)JN G ' JNP.

Now we wish to consider the invariants not with respect to the group JNG,but with respect to the subgroup JNG

(1) ⊂ JNG. We again have

(FunJNg∗)JN G(1)= (Fun JNg∗reg)

JN G(1),

so we will focus on the latter algebra.Let us assume first that N is finite.We show first that the homomorphism (9.6.18) is injective. Let

JNPreg = JNP ×P

g∗reg.


Consider the morphism

JNpreg : JNg∗reg → JNPreg,

A(t) 7→ (JNpreg(A(t)), A(0)).

We claim that it is surjective. Indeed, suppose that we are given a point(φ, x) ∈ JNPreg. Since the map JNpreg is surjective, there exists A(t) ∈ JNg∗regsuch that JNpreg(A(t)) = φ. But then A(0) and x are elements of g∗reg suchthat preg(A(0)) = preg(x). We know that the group G acts transitively alongthe fibers of the map preg. Therefore x = g · A(0) for some g ∈ G. HenceJNpreg(g ·A(t)) = (φ, x). Thus, JNpreg is surjective, and therefore (9.6.18) isinjective.

The group JNG(1) acts along the fibers of the map JNpreg. We claim that

each fiber is a single JNG(1)-orbit.

In order to prove this, we describe the stabilizers of the points of JNg∗regunder the action of the group JNG. Any element of JNG may be written inthe form

g(t) = g0 exp

(N∑

m=1

ymtm

), g0 ∈ G, ym ∈ g.

The condition that g(t) stabilizes a point

x(t) =N∑

n=0

xntn, x0 ∈ g∗reg;xn ∈ g∗, n > 0,

of JNg∗reg is equivalent to a system of recurrence relations on g0 and y1, . . . , ym.The first of these relations is g0 ·x0 = x0, which means that g0 belongs to thestabilizer of x0. The next relation is

ad∗x0(y1) = x1 − g−1

0 · x1.

It is easy to see that for a regular element x0 and any x1 ∈ g∗ the right handside is always in the image of ad∗x0

. Therefore this relation may be resolvedfor y1. But since the centralizer of x0 has dimension ` = rank(g), we obtainan `-dimensional family of solutions for y1.

We continue solving these relations by induction. The mth recurrence re-lation has the form: ad∗x0

(ym) equals an element of g∗ that is completelydetermined by the solutions of the previous relations. If this element of g∗ isin the image of ad∗x0

, then the mth relation may be resolved for ym, and weobtain an `-dimensional family of solutions. Otherwise, there are no solutions,and the recurrence procedure stops. This means that there is no g(t) ∈ JNG

with g(0) = g0 in the stabilizer of x(t).It follows that the maximal possible dimension of the centralizer of any

x(t) ∈ JNg∗reg in JNG is equal to (N + 1)`. This bound is realized if and onlyif for generic g0 ∈ Stabx0(G) all recurrence relations may be resolved (i.e., the


right hand side of themth relation is in the image of ad∗x0for allm = 1, . . . , N).

But if these relations may be resolved for generic g0, then they may be resolvedfor all g0 ∈ Stabx0(G). Thus, we obtain that the centralizer of x(t) ∈ JNg∗regin JNG has dimension (N + 1)` if and only if for any g0 ∈ Stabx0(G) allrecurrence relations may be resolved.

However, observe that the dimension of JNg∗reg is equal to (N + 1) dim g,and the dimension of JNP is equal to (N + 1)`. Therefore the dimension ofa generic fiber of the map JNpreg is equal to (N + 1)(dim g − `). Since thegroup JNG acts transitively along the fibers, we find that the dimension ofthe stabilizer of a generic point of JNg

∗reg is equal to (N +1)`. Therefore for a

generic x(t) ∈ JNg∗reg all of the above recurrence relations may be resolved, forany g0 ∈ Stabx0 . But then they may be resolved for all x(t) and g0 ∈ Stabx0 .In particular, we find that for any x(t) ∈ JNg∗reg and g0 ∈ Stabx0(G) thereexists g(t) ∈ JNG such that g(0) = g0 and g(t) · x(t) = x(t).

Now we are ready to prove that each fiber of JNpreg is a single JNG(1)-

orbit. Consider two points x1(t), x2(t) ∈ JNg∗reg that belong to the same fiberof the map JNpreg. In particular, we have x1(0) = x2(0) = x0. These twopoints also belong to the same fiber of the map JNpreg. Therefore there existsg(t) ∈ JNG such that x2(t) = g(t) · x1(t). This means, in particular, thatg(0) · x0 = x0. According to the previous discussion, there exists g(t) ∈ JNG

such that g(0) = g(0) and g(t) · x2(t) = x2(t). Therefore g(t)−1g(t) ∈ JNG(1)

has the desired property: it transforms x1(t) to x2(t).Thus, we obtain that each fiber of JNpreg is a single JNG

(1)-orbit. In thesame way as in the proof of Theorem 3.4.2 this implies that the algebra ofJNG

(1)-invariants in FunJNg∗reg, and hence in FunJNg∗, is equal to the imageof Fun JNP under the injective homomorphism (9.6.18). This completes theproof for finite values of N .

Finally, consider the case of infinite jet schemes. As before, we will denoteJ∞X by JX. By definition, any element A of Fun JP comes by pull-back froman element AN of Fun JP for sufficiently large N . The image of A in FunJg∗

is the pull-back of the image of AN in Fun JNg∗. Therefore the injectivity ofthe homomorphism (9.6.18) for finite N implies its injectivity for N =∞.

Any JG(1)-invariant function on Jg∗reg comes by pull-back from a JNG(1)-

invariant function on JNg∗reg. Hence we obtain that the algebra of JG(1)-invariant functions on Jg∗reg is equal to Fun JP.

It is useful to note the following straightforward generalization of this propo-sition.

For N as above, let k be a non-negative integer less than N . Denote by

J(k)N P the fiber product

J(k)N P = JNP ×

JkPJkg∗


with respect to the natural morphisms JNP → JkP and Jkg∗ → JkP.Consider the morphism

J(k)N p : JNg∗ → J

(k)N P,

A(t) 7→ (JNp(A(t)), A(t) mod (tk+1))

and the corresponding homomorphism of algebras

Fun J (k)N P → FunJNg∗. (9.6.19)

Let JNG(k+1) be the subgroup of JNG of elements congruent to the identity

modulo (tk+1). Note that the Lie algebra of JNG(k+1) is

JNg(k+1) = g⊗ (tk+1C[[t]]/tN+1C[[t]]).

By repeating the argument used in the proof of Proposition 9.6.4, we obtainthe following generalization of Proposition 9.6.4 (which corresponds to thespecial case k = 0):

Proposition 9.6.5 The homomorphism (9.6.19) is injective and its image isequal to the algebra of JNg(k+1)-invariants in FunJNg∗ (for both finite andinfinite N).

9.6.5 The invariant subspace in the associated graded space

The group G and the Lie algebra g naturally act on the space of JG(1)-invariant polynomial functions on Jg∗ = g∗[[t]], preserving the natural Z+-grading on the algebra Fun g∗[[t]] defined by the degree of the polynomial.By Proposition 9.6.4, this space of JG(1)-invariant functions is isomorphic toFun JP. As a Z+-graded g-module, the latter is isomorphic to

Fun g∗/(Inv g∗)+ ⊗C

C[P i,n]i=1,...,`;n<0.

Here (Inv g∗)+ denotes the ideal in Fun g∗ generated by the augmentationideal of the subalgebra Inv g∗ of g-invariants. The action of g is non-trivialonly on the first factor, and the Z+-grading is obtained by combining thegradings on both factors.

Therefore we have an isomorphism of Z+-graded vector spaces

(Vλ ⊗ V ∗λ ⊗ Fun g∗[[t]])g[[t]]

= Homg(Vλ ⊗ V ∗λ ,Fun g∗/(Inv g∗)+)⊗C

C[P i,n]i=1,...,`;n<0. (9.6.20)

Let us compute the character of (9.6.20).According to a theorem of B. Kostant [Ko], for any g-module V the Z+-

graded space Homg(V,Fun g∗/(Inv g∗)+) is isomorphic to the space of ge-invariants of V , where e is a regular nilpotent element of g and ge is its cen-tralizer (which is an `-dimensional commutative Lie subalgebra of g). Without


loss of generality, we will choose e to be the element∑

ı eı, where eıı=1,...,`

are generators of n. Then e is contained in the sl2-triple f, ρ, e ⊂ g. Thecorresponding Z+-grading on the space V ge is the Z+-grading defined by theaction of ρ with appropriate shift (see below).

Thus, we obtain that the character of the first factor

Homg(Vλ ⊗ V ∗λ ,Fun g∗/(Inv g∗)+) (9.6.21)

in the tensor product appearing in the right hand side of (9.6.20) is equal to

(Vλ ⊗ V ∗λ )ge = HomU(ge)(Vλ, Vλ).

If Vλ were a cyclic module over the polynomial algebra U(ge), generated by alowest weight vector uλ, then any homomorphism of U(ge)-modules Vλ → Vλ

would be uniquely determined by the image of uλ. Hence we would obtainthat

HomU(ge)(Vλ, Vλ) ' Vλ,

where the grading on the right hand side is the principal grading on Vλ minus〈λ, ρ〉 (note that the weight of uλ is equal to −〈λ, ρ〉). Thus, we would findthat the character of (9.6.21) is equal to the principal character of Vλ timesq−〈λ,ρ〉. According to formula (10.9.4) of [K2], the latter is equal to

chVλ =∏

α∈∆+

1− q〈α,λ+ρ〉

1− q〈α,ρ〉 . (9.6.22)

Therefore the sought-after character of (Vλ ⊗ V ∗λ ⊗ Fun g∗[[t]])g[[t]], which isequal to the character of the right hand side of (9.6.20), would be given byformula (9.6.16).

However, Vλ is a cyclic module over U(ge) if and only if λ is a minuscule(i.e., each weight in Vλ has multiplicity one). The “only if” part follows fromthe fact that the dimension of (Vλ ⊗ V ∗λ )ge is equal to the dimension of thesubspace of Vλ ⊗ V ∗λ of weight 0 (see [Ko]), which is equal to∑

µ

(dimVλ(µ))2 >∑

µ

dimVλ(µ) = dimVλ,

unless λ is minuscule (here Vλ(µ) denotes the weight µ component of Vλ).The “if” part is a result of V. Ginzburg, see [G1], Prop. 1.8.1 (see also [G2]).

Thus, for a minuscule weight λ the above calculation implies that the char-acter of Endbgκc

Vλ = (Vλ ⊗ V ∗λ )g[[t]], whose associated graded is contained inthe space (Vλ ⊗ V ∗λ ⊗ Fun g∗[[t]])g[[t]], is less than or equal to the character ofFunOpλ

LG given by formula (9.6.16). Since we already know from the resultsof Section 9.6.2 that Fun Opλ

LG is isomorphic to a subquotient of EndbgκcVλ,

this implies the statement of Theorem 9.6.1 for minuscule highest weights λ.But for a non-minuscule highest weight λ we find that the character of


(Vλ ⊗ V ∗λ ⊗ Fun g∗[[t]])g[[t]] is strictly greater than the character of FunOpλLG,

and hence the above computation cannot be used to prove Theorem 9.6.1.(Furthermore, this implies that the map (9.6.17) is an isomorphism if andonly if λ is minuscule.) We will give another argument in the next section.

9.6.6 Completion of the proof

We will complete the proof of Theorem 9.6.1 by showing that the top hori-zontal homomorphism of the diagram (9.6.8),

(Vλ ⊗ V ∗λ )g[[t]] = (Vλ)bn+λ → FunConn(Ω−ρ)λ

D, (9.6.23)

is injective and its image is contained in the intersection of the kernels of theoperators Vi[λi + 1], i = 1, . . . , `, given by formula (9.6.15).

The following statement (generalizing Proposition 7.1.1, corresponding tothe case λ = 0) is proved in [FG6].

Theorem 9.6.6 The map ıλ : Vλ →Wλ is injective for any dominant integralweight λ.

Therefore the map (9.6.23) is injective. By Lemma 9.5.2, (Wλ)bn+λ is the

subspace

FunConn(Ω−ρ)λD = πλ ⊂Wλ.

Therefore we obtain an injective map

(V)bn+λ → Funπλ. (9.6.24)

Proposition 9.6.7 The image of the map (9.6.24) is contained in the inter-section of the kernels of the operators Vi[λi +1], i = 1, . . . , `, given by formula(9.6.15).

Proof. We use the same proof as in Proposition 7.3.6, which correspondsto the case λ = 0. In Section 7.3.5 we constructed, for any dominant integralweight λ, the screening operators

S(λi)

i : Wλ,κc → W0,λ,κc ,

which commute with the action of gκc . It is clear from formula (7.3.17) forthese operators that they annihilate the highest weight vector in Wλ,κc

. Sincethe image of Vλ in Wλ = Wλ,κc

is generated by this vector, we find that this

image is contained in the intersection of the kernels of the operators S(λi)

i . Butthe image of (Vλ)bn+

λ is also contained in the subspace πλ ⊂ Wλ. Thereforeit is contained in the intersection of the kernels of the operators obtained byresticting S

(λi)

i to πλ ⊂ Wλ. As explained at the end of Section 7.3.5, thisrestriction is equal to the operator V i[λi + 1] given by formula (7.3.18).


Comparing formula (7.3.18) to (9.6.15), we find that upon substitutionbi,n 7→ ui,n the operator V i[λi + 1] becoms the operator Vi[λi + 1] given byformula (9.6.15). This completes the proof.

Now injectivity of the map (9.6.23), Proposition 9.6.7 and Proposition 9.6.3imply that the map

(Vλ)bn+λ → πλ = FunConn(Ω−ρ)λ

D

is injective and its image is contained in Fun OpλLG ⊂ FunConn(Ω−ρ)λ

D. Asexplained in Section 9.6.3 (see the discussion after the diagram (9.6.8)), thisproves Theorem 9.6.1.

10

Constructing the Langlands correspondence

It is time to use the results obtained in the previous chapters in our questfor the local Langlands correspondence for loop groups. So we go back to thequestion posed at the end of Chapter 1: let

LocLG(D×) =∂t +A(t), A(t) ∈ Lg((t))

/ LG((t)) (10.0.1)

be the set of gauge equivalence classes of LG-connections on the punctureddisc D× = Spec C((t)). We had argued in Chapter 1 that LocLG(D×) shouldbe taken as the space of Langlands parameters for the loop group G((t)).Recall that the loop group G((t)) acts on the category gκ -mod of (smooth)g-modules of level κ (see Section 1.3.6 for the definition of this category). Weasked the following question:

Associate to each local Langlands parameter σ ∈ LocLG(D×)a subcategory gκ -modσ of gκ -mod which is stable under theaction of the loop group G((t)).

Even more ambitiously, we wish to represent the category gκ -mod as “fiber-ing” over the space of local Langlands parameters LocLG(D×), with the cat-egories gκ -modσ being the “fibers” and the group G((t)) acting along thesefibers. If we could do that, then we would think of this fibration as a “spectraldecomposition” of the category gκ -mod over LocLG(D×).

At the beginning of Chapter 2 we proposed a possible scenario for solvingthis problem. Namely, we observed that any abelian category may be thoughtof as “fibering” over the spectrum of its center. Hence our idea was to describethe center of the category gκ -mod (for each value of κ) and see if its spectrumis related to the space LocLG(D×) of Langlands parameters.

We have identified the center of the category gκ -mod with the center Zκ(g)of the associative algebra Uκ(g), the completed enveloping algebra of g of levelκ, defined in Section 2.1.2. Next, we described the algebra Zκ(g). Accordingto Proposition 4.3.9, if κ 6= κc, the critical level, then Zκ(g) = C. Thereforeour approach cannot work for κ 6= κc. However, we found that the center

295

296 Constructing the Langlands correspondence

Zκc(g) at the critical level is highly non-trivial and indeed related to LG-connections on the punctured disc.

In this chapter, following my joint works with D. Gaitsgory [FG1]–[FG6],we will use these results to formulate more precise conjectures on the localLanglands correspondence for loop groups and to provide some evidence forthese conjectures. We will also discuss the implications of these conjecturesfor the global geometric Langlands correspondence.

We note that A. Beilinson has another proposal [Bei] for local geometricLanglands correspondence, using representations of affine Kac–Moody alge-bras of levels less than critical. It would be interesting to understand theconnection between his proposal and ours.

Here is a more detailed plan of this chapter. In Section 10.1 we review therelation between local systems and opers. We expect that the true space ofLanglands parameters for loop groups is the space of LG-local systems on thepunctured disc D×. On the other hand, the category of gκc -modules fibersover the space of LG-opers on D×. We will see that this has some non-trivial consequences. Next, in Section 10.2 we introduce the Harish-Chandracategories; these are the categorical analogues of the subspaces of invariantvectors under a compact subgroup K ⊂ G(F ). Our program is to describethe categories of Harish-Chandra modules over gκc in terms of the Langlandsdual group LG and find connections and parallels between the correspond-ing objects in the classical Langlands correspondence. We implement thisprogram in two important examples: the “unramified” case, where our localsystem on D× is isomorphic to the trivial local system, and the “tamely ram-ified” case, where the corresponding connection has regular singularity andunipotent monodromy.

The unramified case is considered in detail in Section 10.3. We first recallthat in the classical setting it is expressed by the Satake correspondence be-tween isomorphism classes of unramified representations of the group G(F )and semi-simple conjugacy classes in the Langlands dual group LG. We thendiscuss the categorical version of this correspondence. We give two realizationsof the categorical representations of the loop group associated to the triviallocal system, one in terms of gκc

-modules and the other in terms of D-moduleson the affine Grassmannian, and the conjectural equivalence between them.In particular, we show that the corresponding category of Harish-Chandramodules is equivalent to the category of vector spaces, in agreement with theclassical picture.

The tamely ramified case is discussed in Section 10.4. In the classical settingthe corresponding irreducible representations of G(F ) are in one-to-one corre-spondence with irreducible modules over the affine Hecke algebra associatedto G(F ). According to [KL, CG], these modules may be realized as subquo-tients of the algebraic K-theory of the Springer fibers associated to unipotentelements of LG (which play the role of the monodromy of the local system in

10.1 Opers vs. local systems 297

the geometric setting). This suggests that in our categorical version the corre-sponding categories of Harish-Chandra modules over gκc

should be equivalentto suitable categories of coherent sheaves on the Springer fibers (more pre-cisely, we should consider here the derived categories). A precise conjecture tothis effect was formulated in [FG2]. In Section 10.4 we discuss this conjectureand various supporting evidence for it, following [FG2]–[FG5]. We note thatthe appearance of the Springer fiber may be traced to the construction asso-ciating Wakimoto modules to Miura opers exposed in the previous chapter.We emphasize the parallels between the classical and the geometric pictures.We also present some explicit calculations in the case of g = sl2, which shouldserve as an illustration of the general theory.

Finally, in Section 10.5 we explain what these results mean for the globalgeometric Langlands correspondence. We show how the localization functorslink local categories of Harish-Chandra modules over gκc

and global categoriesof Hecke eigensheaves on moduli stacks of G-bundles on curves with parabolicstructures. We expect that, at least in the generic situation, the local andglobal categories are equivalent to each others. Therefore our results andconjectures on the local categories translate into explicit statements on thestructure of the categories of Hecke eigensheaves. This way the representationtheory of affine Kac–Moody algebras yields valuable insights into the globalLanglands correspondence.

10.1 Opers vs. local systems

Over the course of several chapters we have given a detailed description ofthe center Zκc

(g) at the critical level. According to Theorem 4.3.6 andrelated results obtained in Chapter 8, Zκc

(g) is isomorphic to the algebraFunOpLG(D×) of functions on the space of LG-opers on the punctured discD×, in a way that is compatible with various symmetries and structures onboth algebras. Now observe that there is a one-to-one correspondence betweenpoints χ ∈ OpLG(D×) and homomorphisms (equivalently, characters)

FunOpLG(D×)→ C,

corresponding to evaluating a function at χ. Hence points of OpLG(D×)parameterize central characters Zκc

(g)→ C.Given a LG-oper χ ∈ OpLG(D×), we define the category

gκc -modχ

as a full subcategory of gκc-mod whose objects are g-modules of critical level

(hence Uκc(g)-modules) on which the center Zκc(g) ⊂ Uκc(g) acts accordingto the central character corresponding to χ. Since the algebra OpLG(D×) actson the category gκc

-mod, we may say that the category gκ -mod “fibers” over


the space OpLG(D×), in such a way that the fiber-category corresponding toχ ∈ OpLG(D×) is the category gκc -modχ.

More generally, for any closed algebraic subvariety Y ⊂ OpLG(D×) (notnecessarily a point), we have an ideal

IY ⊂ FunOpLG(D×) ' Zκc(g)

of those functions that vanish on Y . We then have a full subcategory gκc-modY

of gκc -mod whose objects are g-modules of critical level on which IY acts by0. This category is an example of a “base change” of the category gκc

-modwith respect to the morphism Y → OpLG(D×). It is easy to generalizethis definition to an arbitrary affine scheme Y equipped with a morphismY → OpLG(D×). The corresponding base changed categories gκc -modY maythen be “glued” together, which allows us to define the base changed cate-gory gκc -modY for any scheme Y mapping to OpLG(D×). It is not difficultto generalize this to a general notion of an abelian category fibering over analgebraic stack (see [Ga3]). A model example of a category fibering over analgebraic stack Y is the category of coherent sheaves on an algebraic stack Xequipped with a morphism X → Y.

Recall that the group G((t)) acts on Uκc(g) and on the category gκc -mod.According to Proposition 4.3.8, the action of G((t)) on Zκc

(g) ⊂ Uκc(g) is triv-ial. Therefore the subcategories gκc

-modχ (and, more generally, gκc -modY )are stable under the action of G((t)). Thus, the group G((t)) acts “along thefibers” of the “fibration” gκc -mod → OpLG(D×) (see [FG2], Section 20, formore details).

The fibration gκc -mod → OpLG(D×) almost gives us the desired localLanglands correspondence for loop groups. But there is one important differ-ence: we asked that the category gκc -mod fiber over the space LocLG(D×) oflocal systems on D×. We have shown, however, that gκc

-mod fibers over thespace OpLG(D×) of LG-opers.

What is the difference between the two spaces? While a LG-local systemis a pair (F ,∇), where F is an LG-bundle and ∇ is a connection on F , anLG-oper is a triple (F ,∇,FLB), where F and ∇ are as before, and FLB

is an additional piece of structure; namely, a reduction of F to a (fixed)Borel subgroup LB ⊂ LG satisfying the transversality condition explained inSection 4.2.1. Thus, for any curve X we clearly have a forgetful map

OpLG(X)→ LocLG(X).

The fiber of this map over (F ,∇) ∈ LocLG(X) consists of all LB-reductionsof F satisfying the transversality condition with respect to ∇.

It may well be that this map is not surjective, i.e., that the fiber of thismap over a particular local system (F ,∇) is empty. For example, if X is aprojective curve and LG is a group of adjoint type, then there is a uniqueLG-bundle F0 such that the fiber over (F0,∇) is non-empty. Furthermore,

10.1 Opers vs. local systems 299

for this F0 the fiber over (F0,∇) consists of one point for any connection ∇(in other words, for each connection ∇ on F0 there is a unique LB-reductionFLB of F0 satisfying the transversality condition with respect to ∇).

The situation is quite different when X = D×. In this case any LG-bundleF may be trivialized. A connection ∇ therefore may be represented as a firstorder operator ∂t +A(t), A(t) ∈ Lg((t)). However, the trivialization of F is notunique; two trivializations differ by an element of LG((t)). Therefore the setof equivalence classes of pairs (F ,∇) is identified with the quotient (10.0.1).

Suppose now that (F ,∇) carries an oper reduction FLB . Then we consideronly those trivializations of F which come from trivializations of FLB . Thereare fewer of these, since two trivializations now differ by an element of LB((t))rather than LG((t)). Due to the oper transversality condition, the connection∇must have a special form with respect to any of these trivializations; namely,

∇ = ∂t +∑i=1

ψi(t)fi + v(t),

where each ψi(t) 6= 0 and v(t) ∈ Lb((t)) (see Section 4.2.2). Thus, we obtaina concrete realization of the space of opers as a space of gauge equivalenceclasses

OpLG(D×) =

=

∂t +

∑i=1

ψi(t)fi + v(t), ψi 6= 0,v(t) ∈ Lb((t))

/LB((t)). (10.1.1)

Now the map

α : OpLG(D×)→ LocLG(D×)

simply takes a LB((t))-equivalence class of operators of the form (10.1.1) toits LG((t))-equivalence class.

Unlike the case of projective curves X discussed above, we expect that themap α is surjective for any simple Lie group LG. This has been proved in[De1] in the case of SLn, and we conjecture it to be true in general.

Conjecture 10.1.1 The map α is surjective for any simple Lie group LG.

Now we find ourselves in the following situation: we expect that there existsa category C fibering over the space LocLG(D×) of “true” local Langlandsparameters, equipped with a fiberwise action of the loop group G((t)). Thefiber categories Cσ corresponding to various σ ∈ LocLG(D×) should satisfyvarious, not yet specified, properties. Moreover, we expect C to be the uni-versal category equipped with an action of G((t)). In other words, we expectthat LocLG(D×) is the universal parameter space for the categorical repre-sentations of G((t)) (at the moment we cannot formulate this property more


precisely). The ultimate form of the local Langlands correspondence for loopgroups should be, roughly, the following statement:

categories fiberingover LocLG(D×)

⇐⇒ categories equippedwith action of G((t))

(10.1.2)

The idea is that given a category A on the left hand side we construct acategory on the right hand side by taking the “fiber product” of A and theuniversal category C over LocLG(D×).

Now, we have constructed a category gκc -mod, which fibers over a closecousin of the space LocLG(D×) – namely, the space OpLG(D×) of LG-opers– and is equipped with a fiberwise action of the loop group G((t)).

What should be the relationship between gκc -mod and the conjectural uni-versal category C?

The idea of [FG2] is that the second fibration is a “base change” of the firstone, that is, there is a Cartesian diagram

gκc-mod −−−−→ Cy y

OpLG(D×) α−−−−→ LocLG(D×)

(10.1.3)

that commutes with the action of G((t)) along the fibers of the two verticalmaps. In other words,

gκc -mod ' C ×LocLG(D×)

OpLG(D×).

Thus, gκc -mod should arise as the category on the right hand side of thecorrespondence (10.1.2) attached to the category of quasicoherent sheaveson OpLG(D×) on the left hand side. The latter is a category fibering overLocLG(D×), with respect to the map OpLG(D×)→ LocLG(D×).

At present, we do not have a definition of C, and therefore we cannot makethis isomorphism precise. But we will use it as our guiding principle. Wewill now discuss various corollaries of this conjecture and various pieces ofevidence that make us believe that it is true.

In particular, let us fix a Langlands parameter σ ∈ LocLG(D×) that is in theimage of the map α (according to Conjecture 10.1.1, all Langlands parametersare). Let χ be an LG-oper in the preimage of σ, α−1(σ). Then, according tothe above conjecture, the category gκc -modχ is equivalent to the “would be”Langlands category Cσ attached to σ. Hence we may take gκc -modχ as thedefinition of Cσ.

The caveat is, of course, that we need to ensure that this definition is inde-pendent of the choice of χ in α−1(σ). This means that for any two LG-opers,χ and χ′, in the preimage of σ, the corresponding categories, gκc -modχ andgκc

-modχ′ , should be equivalent to each other, and this equivalence should

10.2 Harish–Chandra categories 301

commute with the action of the loop group G((t)). Moreover, we should ex-pect that these equivalences are compatible with each other as we move alongthe fiber α−1(σ). We will not try to make this condition more precise here(however, we will explain below in Conjecture 10.3.10 what this means forregular opers).

Even putting the questions of compatibility aside, we arrive at the followingrather non-trivial conjecture.

Conjecture 10.1.2 Suppose that χ, χ′ ∈ OpLG(D×) are such that α(χ) =α(χ′), i.e., that the flat LG-bundles on D× underlying the LG-opers χ andχ′ are isomorphic to each other. Then there is an equivalence between thecategories gκc -modχ and gκc -modχ′ which commutes with the actions of thegroup G((t)) on the two categories.

Thus, motivated by our quest for the local Langlands correspondence, wehave found an unexpected symmetry in the structure of the category gκc

-modof g-modules of critical level.

10.2 Harish–Chandra categories

As we explained in Chapter 1, the local Langlands correspondence for theloop group G((t)) should be viewed as a categorification of the local Lang-lands correspondence for the group G(F ), where F is a local non-archimedianfield. This means that the categories Cσ, equipped with an action of G((t)),that we are trying to attach to the Langlands parameters σ ∈ LocLG(D×)should be viewed as categorifications of the smooth representations of G(F )on complex vector spaces attached to the corresponding local Langlands pa-rameters discussed in Section 1.1.5. Here we use the term “categorification”to indicate that we expect the Grothendieck groups of the categories Cσ to“look like” irreducible smooth representations of G(F ).

Our goal in this chapter is to describe the categories Cσ as categories ofgκc

-modules. We begin by taking a closer look at the structure of irreduciblesmooth representations of G(F ).

10.2.1 Spaces of K-invariant vectors

It is known that an irreducible smooth representation (R, π) of G(F ) is auto-matically admissible, in the sense that for any open compact subgroup K,such as the Nth congruence subgroup KN defined in Section 1.1.2, the spaceRπ(K) of K-invariant vectors in R is finite-dimensional. Thus, while most ofthe irreducible smooth representations (R, π) ofG(F ) are infinite-dimensional,they are filtered by the finite-dimensional subspaces Rπ(K) of K-invariant vec-tors, where K are smaller and smaller open compact subgroups. The space


Rπ(K) does not carry an action of G(F ), but it carries an action of the Heckealgebra H(G(F ),K).

By definition, H(G(F ),K) is the space of compactly supported K bi-invariant functions on G(F ). It is given an algebra structure with respectto the convolution product

(f1 ? f2)(g) =∫

G(F )

f1(gh−1)f2(h) dh, (10.2.1)

where dh is the Haar measure on G(F ) normalized in such a way that thevolume of the subgroup K0 = G(O) is equal to 1 (here O is the ring of integersof F ; e.g., for F = Fq((t)) we have O = Fq[[t]]). The algebra H(G(F ),K) actson the space Rπ(K) by the formula

f ? v =∫

G(F )

f1(gh−1)(π(h) · v) dh, v ∈ Rπ(K). (10.2.2)

Studying the spaces of K-invariant vectors and their H(G(F ),K)-modulestructure gives us an effective tool for analyzing representations of the groupG(F ), where F = Fq((t)).

Can we find a similar structure in the categorical local Langlands corre-spondence for loop groups?

10.2.2 Equivariant modules

In the categorical setting a representation (R, π) of the group G(F ) is replacedby a category equipped with an action of G((t)), such as gκc

-modχ. Theopen compact subgroups of G(F ) have obvious analogues for the loop groupG((t)) (although they are, of course, not compact with respect to the usualtopology on G((t))). For instance, we have the “maximal compact subgroup”K0 = G[[t]], or, more generally, the Nth congruence subgroup KN , whoseelements are congruent to 1 modulo tNC[[t]]. Another important example isthe analogue of the Iwahori subgroup. This is the subgroup of G[[t]], whichwe denote by I, whose elements g(t) have the property that their value at 0,that is g(0), belong to a fixed Borel subgroup B ⊂ G.

Now, in the categorical setting, an analogue of a vector in a representationof G(F ) is an object of our category, i.e., a smooth gκc -module (M,ρ), whereρ : gκc

→ EndM . Hence for a subgroup K ⊂ G((t)) of the above type ananalogue of a K-invariant vector in a representation of G(F ) is a smooth gκc

-module (M,ρ) which is stable under the action of K. Recall from Section 1.3.6that for any g ∈ G((t)) we have a new gκc -module (M,ρg), where ρg(x) =ρ(Adg(x)). We say that (M,ρ) is stable under K, or that (M,ρ) is weaklyK-equivariant, if there is a compatible system of isomorphisms between(M,ρ) and (M,ρk) for all k ∈ K. More precisely, this means that for each

10.2 Harish–Chandra categories 303

k ∈ K there exists a linear map TMk : M →M such that

TMk ρ(x)(TM

k )−1 = ρ(Adk(x))

for all x ∈ gκc , and we have

TM1 = IdM , TM

k1TM

k2= TM

k1k2.

Thus, M becomes a representation of the group K.† Consider the corre-sponding representation of the Lie algebra k = LieK on M . Let us assumethat the embedding k → g((t)) lifts to k → gκc (i.e., that the central extensioncocycle is trivial on k). This is true, for instance, for any subgroup containedin K0 = G[[t]], or its conjugate. Then we also have a representation of k onM obtained by restriction of ρ. In general, the two representations do nothave to coincide. If they do coincide, then the module M is called stronglyK-equivariant, or simply K-equivariant.

The pair (gκc,K) is an example of Harish-Chandra pair, that is, a pair

(g,H) consisting of a Lie algebra g and a Lie group H whose Lie algebra iscontained in g. The K-equivariant gκc -modules are therefore called (gκc ,K)Harish-Chandra modules. These are (smooth) gκc

-modules on which theaction of the Lie algebra LieK ⊂ gκc

may be exponentiated to an action of K(we will assume thatK is connected). We denote by gκc

-modK and gκc -modKχ

the full subcategories of gκc-mod and gκc -modχ, respectively, whose objects

are (gκc,K) Harish-Chandra modules.

We will stipulate that the analogues of K-invariant vectors in the categorygκc -modχ are (gκc ,K) Harish-Chandra modules. Thus, while the categoriesgκc

-modχ should be viewed as analogues of smooth irreducible representations(R, π) of the group G(F ), the categories gκc

-modKχ are analogues of the spaces

of K-invariant vectors Rπ(K).Next, we discuss the categorical analogue of the Hecke algebra H(G(F ),K).

10.2.3 Categorical Hecke algebras

We recall that H(G(F ),K) is the algebra of compactly supported K bi-invariant functions on G(F ). We realize it as the algebra of left K-invariantcompactly supported functions on G(F )/K. In Section 1.3.3 we have alreadydiscussed the question of categorification of the algebra of functions on ahomogeneous space like G(F )/K. Our conclusion was that the categoricalanalogue of this algebra, when G(F ) is replaced by the complex loop groupG((t)), is the category of D-modules on G((t))/K. More precisely, this quotienthas the structure of an ind-scheme which is a direct limit of finite-dimensional

† In general, it is reasonable to modify the last condition to allow for a non-trivial two-cocycle and hence a non-trivial central extension of K; however, in the case of interestK does not have any non-trivial central extensions.


algebraic varieties with respect to closed embeddings. The appropriate no-tion of (right) D-modules on such ind-schemes is formulated in [BD1] (see also[FG1, FG2]). As the categorical analogue of the algebra of left K-invariantfunctions on G(F )/K, we take the category H(G((t)),K) of K-equivariant D-modules on the ind-scheme G((t))/K (with respect to the left action of K onG((t))/K). We call it the categorical Hecke algebra associated to K.

It is easy to define the convolution of two objects of the categoryH(G((t)),K)by imitating formula (10.2.1). Namely, we interpret this formula as a compo-sition of the operations of pulling back and integrating functions. Then weapply the same operations to D-modules, thinking of the integral as push-forward. However, here one encounters two problems. The first problem isthat for a general group K the morphisms involved will not be proper, and sowe have to choose between the ∗- and !-push-forward. This problem does notarise, however, if K is such that I ⊂ K ⊂ G[[t]], which will be our main caseof interest. The second, and more serious, issue is that in general the push-forward is not an exact functor, and so the convolution of two D-modules willnot be aD-module, but a complex, more precisely, an object of the correspond-ing K-equivariant (bounded) derived category Db(G((t))/K)K of D-moduleson G((t))/K. We will not spell out the exact definition of this category here,referring the interested reader to [BD1] and [FG2]. The exception is the caseof the subgroup K0 = G[[t]], when the convolution functor is exact and so wemay restrict ourselves to the abelian category of K0-equivariant D-moduleson G((t))/K0.

The category Db(G((t))/K)K has a monoidal structure, and as such it actson the derived category of (gκc

,K) Harish-Chandra modules (again, we referthe reader to [BD1, FG2] for the precise definition). In the special case whenK = K0, we may restrict ourselves to the corresponding abelian categories.This action should be viewed as the categorical analogue of the action ofH(G(F ),K) on the space Rπ(K) of K-invariant vectors discussed above.

Our ultimate goal is understanding the “local Langlands categories” Cσassociated to the “local Langlands parameters” σ ∈ LocLG(D×). We nowhave a candidate for the category Cσ; namely, the category gκc

-modχ, whereσ = α(χ). Therefore gκc

-modχ should be viewed as a categorification of asmooth representation (R, π) of G(F ). The corresponding category gκc

-modKχ

of (gκc ,K) Harish-Chandra modules should therefore be viewed as a categori-fication of Rπ(K). This category is acted upon by the categorical Hecke alge-bra H(G((t)),K). Actually, this action does not preserve the abelian categorygκc -modK

χ ; rather H(G((t)),K) acts on the corresponding derived category.†

† This is the first indication that the local Langlands correspondence should assign to alocal system σ ∈ LocLG(D×) not an abelian, but a triangulated category (equipped withan action of G((t))). We will see in the examples considered below that these triangulatedcategories carry different t-structures. In other words, they may be interpreted as derivedcategories of different abelian categories, and we expect that one of them is bgκc -modχ.

10.3 The unramified case 305

We summarize this analogy in the following table.

Classical theory Geometric theory

Representation of G(F ) Representation of G((t))on a vector space R on a category gκc

-modχ

A vector in R An object of gκc -modχ

The subspace Rπ(K) of The subcategory gκc -modKχ of

K-invariant vectors of R (gκc,K) Harish-Chandra modules

Hecke algebra H(G(F ),K) Categorical Hecke algebra H(G((t)),K)acts on Rπ(K) acts on gκc

-modKχ

Now we may test our proposal for the local Langlands correspondence bystudying the categories gκc -modK

χ of Harish-Chandra modules and comparingtheir structure to the structure of the spaces Rπ(K) of K-invariant vectors ofsmooth representations of G(F ) in the known cases. Another possibility isto test Conjecture 10.1.2 when applied to the categories of Harish-Chandramodules.

In the next section we consider the case of the “maximal compact subgroup”K0 = G[[t]]. We will show that the structure of the categories gκc -modK0

χ

is compatible with the classical results about unramified representations ofG(F ). We then take up the more complicated case of the Iwahori subgroupI in Section 10.4. Here we will also find the conjectures and results of [FG2]to be consistent with the known results about representations of G(F ) withIwahori fixed vectors.

10.3 The unramified case

We first take up the case of the “maximal compact subgroup” K0 = G[[t]]of G((t)) and consider the categories gκc

-modχ which contain non-trivial K0-equivariant objects.

10.3.1 Unramified representations of G(F )

These categories are the analogues of smooth representations of the groupG(F ), where F is a local non-archimedian field (such as Fq((t))), that containnon-zero K0-invariant vectors. Such representations are called unramified.The classification of the irreducible unramified representations of G(F ) is thesimplest case of the local Langlands correspondence discussed in Sections 1.1.4and 1.1.5. Namely, we have a bijection between the sets of equivalence classes


of the following objects:

unramified admissiblehomomorphisms W ′F → LG

⇐⇒ irreducible unramifiedrepresentations of G(F )

(10.3.1)

where W ′F is the Weil–Deligne group introduced in Section 1.1.3.By definition, unramified homomorphisms W ′F −→ LG are those which

factor through the quotient

W ′F →WF → Z

(see Section 1.1.3 for the definitions of these groups and homomorphisms).Its admissibility means that its image in LG consists of semi-simple elements.Therefore the set on the left hand side of (10.3.1) is just the set of conjugacyclasses of semi-simple elements of LG. Thus, the above bijection may bereinterpreted as follows:

semi-simple conjugacyclasses in LG

⇐⇒ irreducible unramifiedrepresentations of G(F )

(10.3.2)

To construct this bijection, we look at the spherical Hecke algebraH(G(F ),K0). According to the Satake isomorphism [Sa], in the interpre-tation of Langlands [L], this algebra is commutative and isomorphic to therepresentation ring of the Langlands dual group LG:

H(G(F ),K0) ' Rep LG. (10.3.3)

We recall that Rep LG consists of finite linear combinations∑

i ai[Vi], wherethe Vi are finite-dimensional representations of LG (without loss of generalitywe may assume that they are irreducible) and ai ∈ C, with respect to themultiplication

[V ] · [W ] = [V ⊗W ].

Because Rep LG is commutative, its irreducible modules are all one-dimensional.They correspond to characters Rep LG→ C. We have a bijection

semi-simple conjugacyclasses in LG

⇐⇒ charactersof Rep LG

(10.3.4)

where the character φγ corresponding to the conjugacy class γ is given by theformula†

φγ : [V ] 7→ Tr(γ, V ).

Now, if (R, π) is a representation of G(F ), then the space Rπ(K0) of K0-invariant vectors in V is a module over H(G(F ),K0). It is easy to show

† It is customary to multiply the right hand side of this formula, for irreducible representa-tion V , by a scalar depending on q and the highest weight of V , but this is not essentialfor our discussion.


that this sets up a one-to-one correspondence between equivalence classes ofirreducible unramified representations of G(F ) and irreducible H(G(F ),K0)-modules. Combining this with the bijection (10.3.4) and the isomorphism(10.3.3), we obtain the sought-after bijections (10.3.1) and (10.3.2).

In particular, we find that, because the Hecke algebra H(G(F ),K0) is com-mutative, the space Rπ(K0) of K0-invariants of an irreducible representation,which is an irreducible H(G(F ),K0)-module, is either 0 or one-dimensional.If it is one-dimensional, then H(G(F ),K0) acts on it by the character φγ forsome γ:

HV ? v = φγ([V ])v = Tr(γ, V )v, v ∈ Rπ(K0), [V ] ∈ Rep LG, (10.3.5)

where HV is the element of H(G(F ),K0) corresponding to [V ] under theisomorphism (10.3.3) (see formula (10.2.2) for the definition of the convolutionaction). Thus, any v ∈ Rπ(K0) is a Hecke eigenvector.

We now discuss the categorical analogues of these statements.

10.3.2 Unramified categories of gκc-modules

In the categorical setting, the role of an irreducible representation (R, π) ofG(F ) is played by the category gκc -modχ for some χ ∈ OpLG(D×). The ana-logue of an unramified representation is a category gκc

-modχ which containsnon-zero (gκc

, G[[t]]) Harish-Chandra modules. This leads us to the follow-ing question: for what χ ∈ OpLG(D×) does the category gκc -modχ containnon-zero (gκc

, G[[t]]) Harish-Chandra modules?We saw in the previous section that (R, π) is unramified if and only if it

corresponds to an unramified Langlands parameter, which is a homomorphismW ′F → LG that factors through W ′F → Z. Recall that in the geometric settingthe Langlands parameters are LG-local systems on D×. The analogues ofunramified homomorphisms W ′F → LG are those local systems on D× whichextend to the disc D, in other words, have no singularity at the origin 0 ∈ D.Note that there is a unique, up to isomorphism local system on D. Indeed,suppose that we are given a regular connection on a LG-bundle F on D. Letus trivialize the fiber F0 of F at 0 ∈ D. Then, because D is contractible, theconnection identifies F with the trivial bundle on D. Under this identificationthe connection itself becomes trivial, i.e., represented by the operator ∇ = ∂t.

Therefore all regular LG-local systems (i.e., those which extend to D) cor-respond to a single point of the set LocLG(D×); namely, the equivalence classof the trivial local system σ0.† From the point of view of the realization ofLocLG(D×) as the quotient (1.2.7) this simply means that there is a unique

† Note however that the trivial LG-local system on D has a non-trivial group of automor-phisms, namely, the group LG itself (it may be realized as the group of automorphismsof the fiber at 0 ∈ D). Therefore if we think of LocLG(D×) as a stack rather than as a

set, then the trivial local system corresponds to a substack pt /LG.


LG((t)) gauge equivalence class containing all regular connections of the form∂t +A(t), where A(t) ∈ Lg[[t]].

The gauge equivalence class of regular connections is the unique local Lang-lands parameter that we may view as unramified in the geometric setting.Therefore, by analogy with the unramified Langlands correspondence forG(F ), we expect that the category gκc -modχ contains non-zero (gκc , G[[t]])Harish-Chandra modules if and only if the LG-oper χ ∈ OpLG(D×) is LG((t))gauge equivalent to the trivial connection, or, in other words, χ belongs tothe fiber α−1(σ0) over σ0.

What does this fiber look like? Let P+ be the set of dominant integralweights of G (equivalently, dominant integral coweights of LG). In Sec-tion 9.2.3 we defined, for each λ ∈ P+, the space Opλ

LG of LB[[t]]-equivalenceclasses of operators of the form

∇ = ∂t +∑i=1

t〈αi,λ〉ψi(t)fi + v(t), (10.3.6)

where ψi(t) ∈ C[[t]], ψi(0) 6= 0, v(t) ∈ Lb[[t]].

Lemma 10.3.1 Suppose that the local system which underlies an oper χ ∈OpLG(D×) is trivial. Then χ belongs to the disjoint union of the subsetsOpλ

LG ⊂ OpLG(D×), λ ∈ P+.

Proof. It is clear from the definition that any oper in OpλLG is regular on

the disc D and is therefore LG((t)) gauge equivalent to the trivial connection.Now suppose that we have an oper χ = (F ,∇,FLB) such that the underly-

ing LG-local system is trivial. Then ∇ is LG((t)) gauge equivalent to a regularconnection, that is one of the form ∂t + A(t), where A(t) ∈ Lg[[t]]. We havethe decomposition LG((t)) = LG[[t]]LB((t)). The gauge action of LG[[t]] clearlypreserves the space of regular connections. Therefore if an oper connection ∇is LG((t)) gauge equivalent to a regular connection, then its LB((t)) gauge classalready must contain a regular connection. The oper condition then impliesthat this gauge class contains a connection operator of the form (10.3.6) forsome dominant integral weight λ of LG. Therefore χ ∈ Opλ

LG.

Thus, we see that the set of opers corresponding to the (unique) unramifiedLanglands parameter is the disjoint union

⊔λ∈P+ Opλ

LG. We call such opers“unramified.” The following result then confirms our expectation that thecategory gκc

-modχ corresponding to an unramified oper χ is also “unrami-fied,” that is, contains non-zero G[[t]]-equivariant objects, if and only if χ isunramified.

Lemma 10.3.2 The category gκc -modχ contains a non-zero


(gκc , G[[t]]) Harish-Chandra module if and only if

χ ∈⊔

λ∈P+

OpλLG . (10.3.7)

Proof. Let us write G[[t]] = G n K1, where K1 is the pro-unipotentpro-algebraic group, which is the first congruence subgroup of G[[t]]. A gκc

-module M is G[[t]]-equivariant if and only if the action of the Lie algebrag ⊗ tC[[t]] of K1 on M is locally nilpotent and exponentiates to an actionof K1 and in addition M decomposes into a direct sum of finite-dimensionalrepresentations under the action of the Lie subalgebra g⊗ 1.

Suppose now that the category gκc-modχ contains a non-zero (gκc , G[[t]])

Harish-Chandra module M . Since M is smooth, any vector v ∈ M is anni-hilated by g ⊗ tNC[[t]] for some N > 0. Consider the g ⊗ tC[[t]]-submoduleMv = U(g ⊗ tC[[t]])v of M . By construction, the action of g ⊗ tC[[t]] onMv factors through the quotient (g⊗ tC[[t]])/(g⊗ tNC[[t]]). Since M is G[[t]]-equivariant, it follows that the action of this Lie algebra may be exponentiatedto an algebraic action of the corresponding algebraic group K1/KN , whereKN is the Nth congruence subgroup of G[[t]], which is a normal subgroup ofK1. The group K1/KN is a finite-dimensional unipotent algebraic group, andwe obtain an algebraic representations of K1/KN on Mv. Any such represen-tation contains an invariant vector.† Thus, we obtain that Mv, and hence M ,contains a vector v′ annihilated by g⊗ tC[[t]].

The g[[t]]-submodule (equivalently, g-submodule) of M generated by v′ is adirect sum of irreducible finite-dimensional representations of g. Let us pickone of those irreducible representations and denote it by M ′. By construction,M ′ is annihilated by g ⊗ tC[[t]] and it is isomorphic to Vλ, the irreduciblerepresentation of g with highest weight λ, as a g-module. Therefore thereexists a non-zero homomorphism Vλ → M , where Vλ is the Weyl moduleintroduced in Section 9.6, sending the generating subspace Vλ ⊂ Vλ to M ′.

By our assumption, M is an object of the category gκc -modχ on whichthe center Z(g) acts according to the central character corresponding to χ ∈OpLG(D×). The existence of a non-zero homomorphism Vλ → M impliesthat χ belongs to the spectrum of the image of Z(g) in Endbgκc

Vλ. Accordingto Theorem 9.6.1, the latter spectrum is equal to Opλ

LG. Therefore we obtainthat χ ∈ Opλ

LG.On the other hand, if χ ∈ Opλ

LG, then the quotient of Vλ by the centralcharacter corresponding to χ is non-zero, by Theorem 9.6.1. This quotient isclearly a G[[t]]-equivariant object of the category gκc -modχ. This completesthe proof.

The next question is to describe the category gκc -modG[[t]]χ of (gκc , G[[t]])

Harish-Chandra modules for χ ∈ OpλLG.

† This follows by induction from the analogous statement for the additive group Ga.


10.3.3 Categories of G[[t]]-equivariant modules

Let us recall from Section 10.3.1 that the space of K0-invariant vectors inan unramified irreducible representation of G(F ) is always one-dimensional.According to our proposal, the category gκc

-modG[[t]]χ should be viewed as a

categorical analogue of this space. Therefore we expect it to be the simplestpossible abelian category: the category of C-vector spaces. Recall that herewe assume that χ belongs to the union of the spaces Opλ

LG, where λ ∈ P+,for otherwise the category gκc

-modG[[t]]χ would be trivial (i.e., the zero object

would be the only object).In this subsection we will prove, following [FG1] (see also [BD1]), that our

expectation is in fact correct provided that λ = 0, in which case Op0LG =

OpLG(D), and so

χ ∈ OpLG(D) ⊂ OpLG(D×).

We will also conjecture that this is true if χ ∈ OpλLG for all λ ∈ P+.

Recall the vacuum module V0 = Vκc(g). According to the isomorphism(9.5.1), we have

EndbgκcV0 ' FunOpLG(D). (10.3.8)

Let χ ∈ OpLG(D) ⊂ OpLG(D×). Then χ defines a character of the algebraEndbgκc

V0. Let V0(χ) be the quotient of V0 by the kernel of this character.Then we have the following result.

Theorem 10.3.3 Let χ ∈ OpLG(D) ⊂ OpLG(D×). Then the categorygκc

-modG[[t]]χ is equivalent to the category of vector spaces: its unique, up to

isomorphism, irreducible object is V0(χ) and any other object is isomorphicto a direct sum of copies of V0(χ).

This theorem may be viewed as a categorical analogue of the local unram-ified Langlands correspondence discussed in Section 10.3.1. It also providesthe first piece of evidence for Conjecture 10.1.2: we see that the categoriesgκc -modG[[t]]

χ are equivalent to each other for all χ ∈ OpLG(D).It is more convenient to consider, instead of an individual regular LG-oper

χ, the entire family Op0LG = OpLG(D) of regular opers on the disc D. Let

gκc-modreg be the full subcategory of the category gκc -mod whose objects

are gκc-modules on which the action of the center Z(g) factors through the

homomorphism

Z(g) ' FunOpLG(D×)→ FunOpLG(D).

Note that the category gκc-modreg is an example of a category gκc -modY

introduced in Section 10.1, in the case when Y = OpLG(D).Let gκc -modG[[t]]

reg be the corresponding G[[t]]-equivariant category. It isinstructive to think of gκc -modreg and gκc -modG[[t]]

reg as categories fibered


over OpLG(D), with the fibers over χ ∈ OpLG(D) being gκc -modχ andgκc

-modG[[t]]χ , respectively.

We will now describe the category gκc -modG[[t]]reg . This description will in

particular imply Theorem 10.3.3.In order to simplify our formulas, in what follows we will use the following

notation for FunOpLG(D):

z = z(g) = FunOpLG(D).

Let z -mod be the category of modules over the commutative algebra z.Equivalently, this is the category of quasicoherent sheaves on OpLG(D).

By definition, any object of gκc -modG[[t]]reg is a z-module. Introduce the

functors

F :gκc -modG[[t]]reg → z -mod, M 7→ Hombgκc

(V0,M),

G :z -mod→ gκc -modG[[t]]reg , F 7→ V0 ⊗

zF .

The following theorem has been proved in [FG1], Theorem 6.3 (importantresults in this direction were obtained earlier in [BD1]).

Theorem 10.3.4 The functors F and G are mutually inverse equivalences ofcategories

gκc-modG[[t]]

reg ' z -mod . (10.3.9)

We will present the proof in the next section. Before doing this, let us notethat it immediately implies Theorem 10.3.3. Indeed, for each χ ∈ OpLG(D)the category gκc

-modG[[t]]χ is the full subcategory of gκc -modG[[t]]

reg whose ob-jects are the gκc

-modules which are annihilated, as z-modules, by the maximalideal Iχ of χ. By Theorem 10.3.4, this category is equivalent to the category ofz-modules annihilated by Iχ. But this is the category of z-modules supported(scheme-theoretically) at the point χ, which is equivalent to the category ofvector spaces.

10.3.4 Proof of Theorem 10.3.4

First of all, we note that the functor F is faithful, i.e., if M 6= 0, then F(M)is non-zero. Indeed, note that

F(M) = Hombgκc(V0,M) = Mg[[t]],

because any gκc -homomorphism V0 → M is uniquely determined by the im-age in M of the generating vector of V0, which is g[[t]]-invariant. As shownin the proof of Lemma 10.3.2, any non-zero gκc

-module M in gκc -modG[[t]]reg

necessarily contains a non-zero vector annihilated by g⊗tC[[t]]. The g-module


generated by this vector has to be trivial, for otherwise the action of the cen-ter on it would not factor through z. Hence M contains a non-zero vectorannihilated by g[[t]], and so Mg[[t]] 6= 0.

Next, we observe that the functor G is left adjoint to the functor F, i.e., wehave a compatible system of isomorphisms

Hom(F ,F(M)) ' Hom(G(F),M). (10.3.10)

In particular, taking M = G(F) in this formula, we obtain a compatiblesystem of maps

F → F G(F),

i.e.,

F → Hombgκc

(V0,V0 ⊗

zF). (10.3.11)

We claim that this is in fact an isomorphism. For this we use the followingtwo results.

Theorem 10.3.5 The z-module V0 is free.

Proof. Recall that we have the PBW filtration on V0, and the associatedgraded is isomorphic to Fun g∗[[t]] (see Section 3.3.3). The PBW filtrationinduces a filtration on z(g) = (V0)g[[t]], and the associated graded is isomor-phic to Inv g∗[[t]] = (Fun g∗[[t]])g[[t]] (see Proposition 4.3.3). Now, accordingto [EF], Fun g∗[[t]] is a free module over Inv g∗[[t]]. This implies in a straight-forward way that the same holds for the original filtered objects, and so V0 isfree over z = FunOpLG(D).

This has the following immediate consequence.

Corollary 10.3.6 The functor G is exact.

Next, let gκc-modG[[t]] be the category of (gκc , G[[t]]) Harish-Chandra mod-

ules (without any restrictions on the action of the center). Let

Extibg,G[[t]](V0,M)

be the higher derived functors of the functor

M 7→ Hombgκc(V0,M)

from gκc -modG[[t]] to the category of vector spaces.

Theorem 10.3.7 ([FT]) We have

Extibg,G[[t]](V0,V0) ' Ωiz, (10.3.12)

the space of differential forms of degree i on OpLG(D).


Note that this is a generalization of the isomorphism

Hombgκc(V0,V0) ' z,

corresponding to the case i = 0 in formula (10.3.12).We will now prove the following isomorphisms:

Extibgκc ,G[[t]] (V0,V0)⊗zF ' Extibgκc ,G[[t]]

(V0,V0 ⊗

zF), i ≥ 0. (10.3.13)

For i = 0 we will then obtain that the map (10.3.11) is an isomorphism, asneeded.

In order to prove the isomorphism (10.3.13), we note that the functors

F 7→ Extibgκc ,G[[t]]

(V0,V0 ⊗

zF)

commute with taking the direct limits. This follows from their realization asthe cohomology of the Chevalley complex computing the relative Lie algebra

cohomology Hi

(g[[t]], g,V0 ⊗

zF)

, as shown in [FT], Prop. 2.1 and Lemma

3.1. Therefore without loss of generality we may, and will, assume that F isa finitely presented z-module.

Since z is isomorphic to a free polynomial algebra (see formula (4.3.1)), anysuch module admits a finite resolution

0→ Pn → . . .→ P1 → P0 → F → 0

by projective modules. According to standard results of homological algebra,the right hand side of (10.3.13) may be computed by the spectral sequence,whose term Ep,q

1 is isomorphic to

Extpbgκc ,G[[t]]

(V0,V0 ⊗

zP−q

).

Since each Pj is projective, we have

Extpbgκc ,G[[t]]

(V0,V0 ⊗

zP−q

)= Extpbgκc ,G[[t]]

(V0,V0)⊗zP−q.

Next, by Theorem 10.3.7, each Extpbgκc ,G[[t]](V0,V0) is a free z-module. There-

fore the second term of the spectral sequence is equal to

Ep,02 = Extpbgκc ,G[[t]]

(V0,V0)⊗zF ,

and Ep,q = 0 for q 6= 0. Therefore the spectral sequence degenerates in thesecond term and the result is the isomorphism (10.3.13).

Thus, we have proved that F G ' Id. Let us prove that G F ' Id.Note that by adjunction (10.3.10) we have a map G F → Id, i.e., a com-

patible system of maps

G F(M)→M, (10.3.14)


for M in gκc -modG[[t]]reg . We need to show that this map is an isomorphism.

Let us show that the map (10.3.14) is injective. If M ′ is the kernel ofG F(M)→M , then by the left exactness of F, we would obtain that

F(M ′) = Ker(F G F(M)→ F(M)

)' Ker

(F(M)→ F(M)

)= 0.

But we know that the functor F is faithful, so M ′ = 0.Next, we show that the map (10.3.14) is surjective. Let M ′′ be the cokernel

of G F(M)→M . We have the long exact sequence

0→ F G F(M)→ F(M)→ F(M ′′)→ R1F(G F(M))→ . . . , (10.3.15)

where R1F is the first right derived functor of the functor F. Thus, we have

R1F(L) = Ext1bgκc -modG[[t]]reg

(V0, L).

To complete the proof, we need the following:

Lemma 10.3.8 For any z-module F we have

Ext1bgκc -modG[[t]]reg

(V0,V0 ⊗

zF)

= 0.

Proof. In the case when F = z this was proved in [BD1]. Here we followthe proof given in [FG1] in the general case.

Any element of

Ext1bgκc -modG[[t]]

(V0,V0 ⊗

zF)

= Ext1bgκc ,G[[t]]

(V0,V0 ⊗

zF)

defines an extension of the gκc -modules

0→ V0 ⊗zF →M→ V0 → 0. (10.3.16)

Thus, M is an extension of two objects of the category gκc -modG[[t]]reg , on

which the center Z(g) acts through the quotient Z(g) → z. But the moduleM may not be an object of gκc

-modG[[t]]reg . The statement of the proposition

means that ifM is an object of gκc-modG[[t]]

reg , then it is necessarily split as agκc

-module.Let I be the ideal of z(g) in Z(g). Then M is an object of gκc -modG[[t]]

reg

if and only if I acts on it by 0. Note that I necessarily vanishes on thesubmodule V0⊗

zF and the quotient V0. Therefore, choosing a linear splitting

of (10.3.16) and applying I to the image of V0 inM, we obtain a map

I → Hombgκc

(V0,V0 ⊗

zF).

Moreover, any element of I2 ⊂ I maps to 0. Thus, we obtain a map

(I/I2)⊗z

Ext1bgκc ,G[[t]]

(V0,V0 ⊗

zF)→ Hombgκc

(V0,V0 ⊗

zF).


We need to show that for any element of Ext1bgκc ,G[[t]]

(V0,V0 ⊗

zF)

the cor-

responding map

I/I2 → Hombgκc

(V0,V0 ⊗

zF)

is non-zero. Equivalently, we need to show that the kernel of the map

Ext1bgκc ,G[[t]]

(V0,V0 ⊗

zF)→ Homz

(I/I2,Hombgκc

(V0,V0 ⊗

zF))

is 0. In fact, this kernel is equal to Ext1bgκc -modG[[t]]reg

(V0,V0 ⊗

zF)

, whose

vanishing we wish to prove.Now, according to Theorem 10.3.7 and formula (10.3.13), the last map may

be rewritten as follows:

Ω1z ⊗

zF → Homz

(I/I2,F

). (10.3.17)

In order to write down an explicit formula for this map, let us explain,following [FT], how to construct an isomorphism

Ω1z ⊗

zF ' Ext1bgκc ,G[[t]]

(V0,V0 ⊗

zF), (10.3.18)

or, in other words, how to construct an extension M of the form (10.3.16)starting from an element of Ω1

z ⊗zF . For simplicity let us suppose that F = z

(the general case is similar). Then as a vector space, the extension M is thedirect sum of two copies of V0. By linearlity, we may assume that our elementof Ω1

z has the form BdA, where A,B ∈ z. Since V0 is generated by the vacuumvector |0〉, the extension M is uniquely determined by the action of g[[t]] onthe vector |0〉 in the summand corresponding to the quotient V0. We then set

X · |0〉 = Bcdot limε→0

1εX ·Aε, X ∈ g[[t]].

Here Aε is an arbitrary deformation of A ∈ z(g) ⊂ V0 = Vκc(g), considered asan element of the vacuum module Vκc+εκ0(g) of level κc + εκ0 (where κ0 is anon-zero invariant inner product on g). Since X · A = 0 in V0 = Vκc

(g) forall X ∈ g[[t]], the right hand side of this formula is well-defined.

Now recall that the center Z(g) is a Poisson algebra with the Poissonbracket defined in Section 8.3.1. Moreover, according to Theorem 8.3.1 andLemma 8.3.2, this Poisson algebra is isomorphic to the Poisson algebra

FunOpG(D×)

with the Poisson algebra structure obtained via the Drinfeld–Sokolov reduc-tion. It is easy to see that with respect to this Poisson structure the ideal I


is Poisson, i.e., I, I ⊂ I. In this situation the Poisson bracket map

FunOpG(D×)→ Θ(OpG(D×)),

where Θ(OpG(D×)) is the space of vector fields on OpG(D×), induces a map

I/I2 → Θz, (10.3.19)

where Θz is the space of vector fields on OpG(D).It follows from the above description of the isomorphism (10.3.18) and the

definition of the Poisson structure on Z(g) from Section 8.3.1 that the map(10.3.17) is the composition of the tautological isomorphism

Ω1z ⊗

zF ∼−→ Homz (Θz,F)

(note that Ω1z is a free z-module) and the map

Homz (Θz,F)→ Homz

(I/I2,F

)(10.3.20)

induced by (10.3.19). But according to [BD1], Theorem 3.6.7, the map(10.3.19) is surjective (in fact, I/I2 is isomorphic to the Lie algebroid onOpLG(D) corresponding to the universal LG-bundle, and (10.3.19) is the cor-responding anchor map, so its kernel is free as a z-module). Therefore weobtain that for any z-module F the map (10.3.20) is injective. Hence the map(10.3.17) is also injective. This completes the proof.

By Lemma 10.3.8, the sequence (10.3.15) gives us a short exact sequence

0→ F G F(M)→ F(M)→ F(M ′′)→ 0.

But the first arrow is an isomorphism because F G ' Id, which implies thatF(M ′′) = 0 and hence M ′′ = 0.

Theorem 10.3.4, and therefore Theorem 10.3.3, are now proved.

10.3.5 The action of the spherical Hecke algebra

In Section 10.3.1 we discussed irreducible unramified representations of thegroup G(F ), where F is a local non-archimedian field. We have seen thatsuch representations are parameterized by conjugacy classes of the Lang-lands dual group LG. Given such a conjugacy class γ, we have an irre-ducible unramified representation (Rγ , πγ), which contains a one-dimensionalsubspace (Rγ)πγ(K0) of K0-invariant vectors. The spherical Hecke algebraH(G(F ),K0), which is isomorphic to Rep LG via the Satake isomorphism,acts on this space by a character φγ , see formula (10.3.5).

In the geometric setting, we have argued that for any χ ∈ OpLG(D) thecategory gκc

-modχ, equipped with an action of the loop group G((t)), shouldbe viewed as a categorification of (Rγ , πγ). Furthermore, its subcategorygκc

-modG[[t]]χ of (gκc , G[[t]]) Harish-Chandra modules should be viewed as


a “categorification” of the one-dimensional space (Rγ)πγ(K0). According toTheorem 10.3.3, the latter category is equivalent to the category of vectorspaces, which is indeed the categorification of a one-dimensional vector space.So this result is consistent with the classical picture.

We now discuss the categorical analogue of the action of the spherical Heckealgebra on this one-dimensional space.

As explained in Section 10.2.3, the categorical analogue of the sphericalHecke algebra is the category of G[[t]]-equivariant D-modules on the affineGrassmannian Gr = G((t))/G[[t]]. We refer the reader to [BD1, FG2] forthe precise definition of Gr and this category. There is an important propertythat is satisfied in the unramified case: the convolution functors with these D-modules are exact, which means that we do not need to consider the derivedcategory; the abelian category of such D-modules will do. Let us denote thisabelian category by H(G((t)), G[[t]]). This is the categorical version of thespherical Hecke algebra H(G((t)), G[[t]]).

According to the results of [MV], the category H(G((t)), G[[t]]) carries anatural structure of tensor category, which is equivalent to the tensor categoryRep LG of representations of LG. This should be viewed as a categoricalanalogue of the Satake isomorphism. Thus, for each object V of Rep LG wehave an object of H(G((t)), G[[t]]) which we denote by HV . What should bethe analogue of the Hecke eigenvector property (10.3.5)?

As we explained in Section 10.2.3, the category H(G((t)), G[[t]]) naturallyacts on the category gκc -modG[[t]]

χ , and this action should be viewed as acategorical analogue of the action of H(G(F ),K0) on (Rγ)πγ(K0).

Now, by Theorem 10.3.3, any object of gκc-modG[[t]]

χ is a direct sum of copiesof V0(χ). Therefore it is sufficient to describe the action of H(G((t)), G[[t]])on V0(χ). This action is described by the following statement, which followsfrom [BD1]: there exists a family of isomorphisms

αV : HV ? V0(χ) ∼−→ V ⊗ V0(χ), V ∈ Rep LG, (10.3.21)

where V is the vector space underlying the representation V (see [FG2, FG4]for more details). Moreover, these isomorphisms are compatible with thetensor product structure on HV (given by the convolution) and on V (givenby tensor product of vector spaces).

In view of Theorem 10.3.3, this is not surprising. Indeed, it follows from thedefinition that HV ? V0(χ) is again an object of the category gκc -modG[[t]]

χ .Therefore it must be isomorphic to UV ⊗C V0(χ), where UV is a vector space.But then we obtain a functor

H(G((t)), G[[t]])→ Vect, HV 7→ UV .

It follows from the construction that this is a tensor functor. Therefore thestandard Tannakian formalism implies that UV is isomorphic to V .

The isomorphisms (10.3.21) should be viewed as the categorical analogues


of the Hecke eigenvector conditions (10.3.5). The difference is that whilein (10.3.5) the action of elements of the Hecke algebra on a K0-invariantvector in Rγ amounts to multiplication by a scalar, the action of an objectof the Hecke category H(G((t)), G[[t]]) on the G[[t]]-equivariant object V0(χ)of gκc -modχ amounts to multiplication by a vector space; namely, the vectorspace underlying the corresponding representation of LG. It is natural to call amodule satisfying this property a Hecke eigenmodule. Thus, we obtain thatV0(χ) is a Hecke eigenmodule. This is in agreement with our expectation thatthe category gκc

-modG[[t]]χ is a categorical version of the space of K0-invariant

vectors in Rγ .One ingredient that is missing in the geometric case is the conjugacy class γ

of LG. We recall that in the classical Langlands correspondence this was theimage of the Frobenius element of the Galois group Gal(Fq/Fq), which doesnot have an analogue in the geometric setting where our ground field is C,which is algebraically closed. So while unramified local systems in the classicalcase are parameterized by the conjugacy classes γ, there is only one, up to anisomorphism, unramified local system in the geometric case. However, thislocal system has a large group of automorphisms; namely, LG itself. We willargue that what replaces γ in the geometric setting is the action of this groupLG by automorphisms of the category gκc

-modχ. We will discuss this in thenext two sections.

10.3.6 Categories of representations and D-modules

When we discussed the procedure of categorification of representations inSection 1.3.4, we saw that there are two possible scenarios for constructingcategories equipped with an action of the loop group G((t)). In the first onewe consider categories of D-modules on the ind-schemes G((t))/K, where Kis a “compact” subgroup of G((t)), such as G[[t]] or the Iwahori subgroup. Inthe second one we consider categories of representations gκc

-modχ. So far wehave focused exclusively on the second scenario, but it is instructive to alsodiscuss categories of the first type.

In the toy model considered in Section 1.3.3 we discussed the category of g-modules with fixed central character and the category ofD-modules on the flagvariety G/B. We have argued that both could be viewed as categorifications ofthe representation of the group G(Fq) on the space of functions on (G/B)(Fq).These categories are equivalent, according to the Beilinson–Bernstein theory,with the functor of global sections connecting the two. Could something likethis be true in the case of affine Kac–Moody algebras as well?

The affine Grassmannian Gr = G((t))/G[[t]] may be viewed as the simplestpossible analogue of the flag variety G/B for the loop group G((t)). Considerthe category of D-modules on G((t))/G[[t]] (see [BD1, FG2] for the precisedefinition). We have a functor of global sections from this category to the


category of g((t))-modules. In order to obtain gκc -modules, we need to takeinstead the category Dκc

-mod of D-modules twisted by a line bundle Lκc.

This is the unique line bundle Lκcon Gr which carries an action of gκc

(suchthat the central element 1 is mapped to the identity) lifting the natural actionof g((t)) on Gr. Then for any object M of Dκc -mod, the space of globalsections Γ(Gr,M) is a gκc

-module. Moreover, it is known (see [BD1, FG1])that Γ(Gr,M) is in fact an object of gκc

-modreg. Therefore we have a functorof global sections

Γ : Dκc-mod→ gκc -modreg .

We note that the categories D -mod and Dκc -mod are equivalent under thefunctor M 7→ M⊗ Lκc . But the corresponding global sections functors arevery different.

However, unlike in the Beilinson–Bernstein scenario, the functor Γ cannotpossibly be an equivalence of categories. There are two reasons for this. Firstof all, the category gκc

-modreg has a large center, namely, the algebra z =FunOpLG(D), while the center of the category Dκc

-mod is trivial.† Thesecond, and more serious, reason is that the category Dκc

-mod carries anadditional symmetry, namely, an action of the tensor category RepLG ofrepresentations of the Langlands dual group LG, and this action trivializesunder the functor Γ as we explain presently.

Over OpLG(D) there exists a canonical principal LG-bundle, which we willdenote by P. By definition, the fiber of P at χ = (F ,∇,FLB) ∈ OpLG(D)is F0, the fiber at 0 ∈ D of the LG-bundle F underlying χ. For an objectV ∈ Rep LG let us denote by V the associated vector bundle over OpLG(D),i.e.,

V = P ×LG

V.

Next, consider the category Dκc -modG[[t]] of G[[t]]-equivariant Dκc -moduleson Gr. It is equivalent to the category

D -modG[[t]] = H(G((t)), G[[t]])

considered above. This is a tensor category, with respect to the convolutionfunctor, which is equivalent to the category Rep LG. We will use the samenotation HV for the object of Dκc -modG[[t]] corresponding to V ∈ Rep LG.The category Dκc

-modG[[t]] acts on Dκc-mod by convolution functors

M 7→ HV ?M

which are exact. This amounts to a tensor action of the category RepLG onDκc -mod.

† Recall that we are under the assumption that G is a connected simply-connected algebraicgroup, and in this case Gr has one connected component. In general, the center of thecategory Dκc -mod has a basis enumerated by the connected components of Gr and isisomorphic to the group algebra of the finite group π1(G).


Now, it follows from the results of [BD1] that there are functorial isomor-phisms

Γ(Gr,HV ?M) ' Γ(Gr,M)⊗zV, V ∈ Rep LG,

compatible with the tensor structure (see [FG2, FG4] for details). Thus, wesee that there are non-isomorphic objects of Dκc -mod, which the functor Γsends to isomorphic objects of gκc

-modreg. Therefore the category Dκc-mod

and the functor Γ need to be modified in order to have a chance to obtain acategory equivalent to gκc -modreg.

In [FG2] it was shown how to modify the category Dκc -mod, by simulta-neously “adding” to it z as a center, and “dividing” it by the above Rep LG-action. As the result, we obtain a candidate for a category that can be equiv-alent to gκc -modreg. This is the category of Hecke eigenmodules on Gr,denoted by DHecke

κc-modreg.

By definition, an object ofDHeckeκc

-modreg is an object ofDκc -mod, equippedwith an action of the algebra z by endomorphisms and a system of isomor-phisms

αV : HV ?M ∼−→ V ⊗zM, V ∈ Rep LG,

compatible with the tensor structure.The above functor Γ naturally gives rise to a functor

ΓHecke : DHeckeκc

-modreg → gκc -modreg . (10.3.22)

This is in fact a general property. To explain this, suppose for simplicitythat we have an abelian category C which is acted upon by the tensor categoryRepH, where H is an algebraic group; we denote this action by

M 7→M ? V, V ∈ RepH.

Let CHecke be the category whose objects are collections (M, αV V ∈Rep H),where M∈ C and αV is a compatible system of isomorphisms

αV :M ? V∼−→ V ⊗

CM, V ∈ RepH,

where V is the vector space underlying V . The category CHecke carries anatural action of the group H: for h ∈ H, we have

h · (M, αV V ∈Rep H) = (M, (h⊗ idM) αV V ∈Rep H).

In other words, M remains unchanged, but the isomorphisms αV get com-posed with h.

The category C may be reconstructed as the category of H-equivariantobjects of CHecke with respect to this action, see [AG]. Therefore one maythink of CHecke as the “de-equivariantization” of the category C.


Suppose that we have a functor G : C → C′, such that we have functorialisomorphisms

G(M ? V ) ' G(M)⊗CV , V ∈ RepH, (10.3.23)

compatible with the tensor structure. Then, according to [AG], there exists afunctor

GHecke : CHecke → C′

such that G ' GHecke Ind, where the functor Ind : C → CHecke sends M toM ? OH , where OH is the regular representation of H. The functor GHecke

may be explicitly described as follows: the isomorphisms αV and (10.3.23)give rise to an action of the algebra OH on G(M), and GHecke(M) is obtainedby taking the fiber of G(M) at 1 ∈ H.

In our case, we take C = Dκc -mod, C′ = gκc -modreg, and G = Γ. The onlydifference is that now we are working over the base OpLG(D), which we haveto take into account. Then the functor Γ factors as

Γ ' ΓHecke Ind,

where ΓHecke is the functor (10.3.22) (see [FG2, FG4] for more details). More-over, the left action of the group G((t)) on Gr gives rise to its action on thecategory DHecke

κc-modreg, and the functor ΓHecke intertwines this action with

the action of G((t)) on gκc-modreg.

The following was conjectured in [FG2]:

Conjecture 10.3.9 The functor ΓHecke in formula (10.3.22) defines an equiv-alence of the categories DHecke

κc-modreg and gκc -modreg.

It was proved in [FG2] that the functor ΓHecke, when extended to the derivedcategories, is fully faithful. Furthermore, it was proved in [FG4] that it setsup an equivalence of the corresponding I0-equivariant categories, where I0 =[I, I] is the radical of the Iwahori subgroup.

Let us specialize Conjecture 10.3.9 to a point χ = (F ,∇,FLB) ∈ OpLG(D).Then on the right hand side we consider the category gκc -modχ, and on theleft hand side we consider the category DHecke

κc-modχ. Its object consists of a

Dκc-module M and a collection of isomorphisms

αV : HV ?M ∼−→ VF0 ⊗M, V ∈ Rep LG. (10.3.24)

Here VF0 is the twist of the representation V by the LG-torsor F0. Theseisomorphisms have to be compatible with the tensor structure on the categoryH(G((t)), G[[t]]).

Conjecture 10.3.9 implies that there is a canonical equivalence of categories

DHeckeκc

-modχ ' gκc -modχ . (10.3.25)


It is this conjectural equivalence that should be viewed as an analogue of theBeilinson–Bernstein equivalence.

From this point of view, we may think of each of the categories DHeckeκc

-modχ

as the second incarnation of the sought-after Langlands category Cσ0 corre-sponding to the trivial LG-local system.

Now we give another explanation why it is natural to view the categoryDHecke

κc-modχ as a categorification of an unramified representation of the group

G(F ). First of all, observe that these categories are all equivalent to each otherand to the category DHecke

κc-mod, whose objects are Dκc -modulesM together

with a collection of isomorphisms

αV : HV ?M ∼−→ V ⊗M, V ∈ Rep LG. (10.3.26)

Comparing formulas (10.3.24) and (10.3.26), we see that there is an equiva-lence

DHeckeκc

-modχ ' DHeckeκc

-mod,

for each choice of trivialization of the LG-torsor F0 (the fiber at 0 ∈ D of theprincipal LG-bundle F on D underlying the oper χ).

Now recall from Section 10.3.1 that to each semi-simple conjugacy class γ inLG corresponds an irreducible unramified representation (Rγ , πγ) of G(F ) viathe Satake correspondence (10.3.2). It is known that there is a non-degeneratepairing

〈, 〉 : Rγ ×Rγ−1 → C,

in other words, Rγ−1 is the representation of G(F ) which is contragredient toRγ (it may be realized in the space of smooth vectors in the dual space toRγ).

Let v ∈ Rγ−1 be a non-zero vector such that K0v = v (this vector is uniqueup to a scalar). It then satisfies the Hecke eigenvector property (10.3.5) (inwhich we need to replace γ by γ−1). This allows us to embed Rγ into the spaceof smooth locally constant right K0-invariant functions on G(F ) (equivalently,functions on G(F )/K0), by using matrix coefficients, as follows:

u ∈ Rγ 7→ fu, fu(g) = 〈u, gv〉.

The Hecke eigenvector property (10.3.5) implies that the functions fu are rightK0-invariant and satisfy the condition

f ? HV = Tr(γ−1, V )f, (10.3.27)

where ? denotes the convolution product (10.2.1). Let C(G(F )/K0)γ be thespace of smooth locally constant functions on G(F )/K0 satisfying (10.3.27).It carries a representation of G(F ) induced by its left action on G(F )/K0. Wehave constructed an injective map Rγ → C(G(R)/G(R))γ , and one can showthat for generic γ it is an isomorphism.


Thus, we obtain a realization of an irreducible unramified representation ofG(F ) in the space of functions on the quotient G(F )/K0 satisfying the Heckeeigenfunction condition (10.3.27). The Hecke eigenmodule condition (10.3.26)may be viewed as a categorical analogue of (10.3.27). Therefore the categoryDHecke

κc-mod of twisted D-modules on Gr = G((t))/K0 satisfying the Hecke

eigenmodule condition (10.3.26), equipped with its G((t))-action, appears tobe a natural categorification of the irreducible unramified representations ofG(F ).

10.3.7 Equivalences between categories of modules

All opers in OpLG(D) correspond to one and the same LG-local system;namely, the trivial local system. Therefore, according to Conjecture 10.1.2,we expect that the categories gκc -modχ are equivalent to each other. Moreprecisely, for each isomorphism between the underlying local systems of anytwo opers in OpLG(D) we wish to have an equivalence of the correspondingcategories, and these equivalences should be compatible with respect to theoperation of composition of these isomorphisms.

Let us spell this out in detail. Let χ = (F ,∇,FLB) and χ′ = (F ′,∇′,F ′LB)be two opers in OpLG(D). Then an isomorphism between the underlyinglocal systems (F ,∇) ∼−→ (F ′,∇′) is the same as an isomorphism F0

∼−→ F ′0between the LG-torsors F0 and F ′0, which are the fibers of the LG-bundlesF and F ′, respectively, at 0 ∈ D. Let us denote this set of isomorphisms byIsomχ,χ′ . Then we have

Isomχ,χ′ = F0 ×LG

LG ×LGF ′0,

where we twist LG by F0 with respect to the left action and by F ′0 withrespect to the right action. In particular,

Isomχ,χ = LGF0 = F0 ×LG

Ad LG

is just the group of automorphisms of F0.It is instructive to combine the sets Isomχ,χ′ into a groupoid Isom over

OpLG(D). Thus, by definition Isom consists of triples (χ, χ′, φ), where χ, χ′ ∈OpLG(D) and φ ∈ Isomχ,χ is an isomorphism of the underlying local systems.The two morphisms Isom→ OpLG(D) correspond to sending such a triple toχ and χ′. The identity morphism OpLG(D)→ Isom sends χ to (χ, χ, Id), andthe composition morphism

Isom ×OpLG(D)

Isom→ Isom

corresponds to composing two isomorphisms.Conjecture 10.1.2 has the following more precise formulation for regular

opers.


Conjecture 10.3.10 For each φ ∈ Isomχ,χ′ there exists an equivalence

Eφ : gκc-modχ → gκc -modχ′ ,

which intertwines the actions of G((t)) on the two categories, such that EId =Id and there exist isomorphisms βφ,φ′ : Eφφ′ ' Eφ Eφ′ satisfying

βφφ′,φ′′βφ,φ′ = βφ,φ′φ′′βφ′,φ′′

for all isomorphisms φ, φ′, φ′′, whenever they may be composed in the appro-priate order.

In other words, the groupoid Isom over OpLG(D) acts on the categorygκc -modreg fibered over OpLG(D), preserving the action of G((t)) along thefibers.

In particular, this conjecture implies that the group LGF0 acts on the cat-egory gκc

-modχ for any χ ∈ OpLG(D).Now we observe that Conjecture 10.3.9 implies Conjecture 10.3.10. Indeed,

by Conjecture 10.3.9, there is a canonical equivalence of categories (10.3.25),

DHeckeκc

-modχ ' gκc -modχ .

It follows from the definition of the category DHeckeκc

-modχ (namely, formula(10.3.24)) that for each isomorphism φ ∈ Isomχ,χ′ , i.e., an isomorphism ofthe LG-torsors F0 and F ′0 underlying the opers χ and χ′, there is a canonicalequivalence

DHeckeκc

-modχ ' DHeckeκc

-modχ′ .

Therefore we obtain the sought-after equivalence

Eφ : gκc -modχ → gκc -modχ′ .

Furthermore, it is clear that these equivalences satisfy the conditions of Con-jecture 10.3.10. In particular, they intertwine the actions of G((t)), because theaction of G((t)) affects the D-moduleM underlying an object of DHecke

κc-modχ,

but does not affect the isomorphisms αV .Equivalently, we can express this by saying that the groupoid Isom naturally

acts on the category DHeckeκc

-modreg. By Conjecture 10.3.9, this gives rise toan action of Isom on gκc

-modreg.In particular, we construct an action of the group (LG)F0 , the twist of

LG by the LG-torsor F0 underlying a particular oper χ, on the categoryDHecke

κc-modχ. Indeed, each element g ∈ (LG)F0 acts on the F0-twist VF0 of

any finite-dimensional representation V of LG. Given an object (M, (αV ))of DHecke

κc-modχ′ , we construct a new object; namely, (M, ((g ⊗ IdM) αV )).

Thus, we do not change the D-moduleM, but we change the isomorphisms αV

appearing in the Hecke eigenmodule condition (10.3.24) by composing themwith the action of g on VF0 . According to Conjecture 10.3.9, the category


DHeckeκc

-modχ is equivalent to gκc -modχ. Therefore this gives rise to an actionof the group (LG)F0 on gκc -modχ. But this action is much more difficult todescribe in terms of gκc

-modules. We will see examples of the action of thesesymmetries below.

10.3.8 Generalization to other dominant integral weights

We have extensively studied above the categories gκc -modχ and gκc -modG[[t]]χ

associated to regular opers χ ∈ OpLG(D). However, by Lemma 10.3.1, the(set-theoretic) fiber of the map α : OpLG(D×)→ LocLG(D×) over the triviallocal system σ0 is the disjoint union of the subsets Opλ

LG, λ ∈ P+. Herewe discuss briefly the categories gκc

-modχ and gκc -modG[[t]]χ for χ ∈ Opλ

LG,where λ 6= 0.

Consider the Weyl module Vλ with highest weight λ defined in Section 9.6.According to Theorem 9.6.1, we have

EndbgκcVλ ' FunOpλ

LG . (10.3.28)

Let χ ∈ OpλLG ⊂ OpLG(D×). Then χ defines a character of the algebra

EndbgκcVλ. Let Vλ(χ) be the quotient of Vλ by the kernel of this character.

The following conjecture of [FG6] is an analogue of Theorem 10.3.3:

Conjecture 10.3.11 Let χ ∈ OpλLG ⊂ OpLG(D×). Then the category

gκc-modG[[t]]

χ is equivalent to the category of vector spaces: its unique, up toisomorphism, irreducible object is Vλ(χ) and any other object is isomorphicto a direct sum of copies of Vλ(χ).

Note that this is consistent with Conjecture 10.1.2, which tells us that thecategories gκc -modG[[t]]

χ should be equivalent to each other for all opers whichare gauge equivalent to the trivial local system on D.

As in the case λ = 0, it is useful to consider, instead of an individual LG-oper χ, the entire family Opλ

LG. Let gκc-modλ,reg be the full subcategory of

the category gκc-mod whose objects are gκc -modules on which the action of

the center Z(g) factors through the homomorphism

Z(g) ' FunOpLG(D×)→ zλ,

where we have set

zλ = FunOpλLG .

Let gκc -modG[[t]]λ,reg be the corresponding G[[t]]-equivariant category. Thus,

gκc -modλ,reg and gκc -modG[[t]]λ,reg are categories fibered over Opλ

LG, with thefibers over χ ∈ Opλ

LG being gκc -modχ and gκc -modG[[t]]χ , respectively.

We have the following conjectural description of the category gκc -modG[[t]]λ,reg ,

which implies Conjecture 10.3.3 (see [FG6]).


Let zλ -mod be the category of modules over the commutative algebra zλ.Equivalently, this is the category of quasicoherent sheaves on the space Opλ

LG.By definition, any object of gκc -modG[[t]]

λ,reg is a zλ-module. Introduce thefunctors

Fλ :gκc -modG[[t]]λ,reg → zλ -mod, M 7→ Hombgκc

(Vλ,M),

Gλ :zλ -mod→ gκc -modG[[t]]λ,reg , F 7→ Vλ ⊗

zλ

F .

Conjecture 10.3.12 The functors Fλ and Gλ are mutually inverse equiva-lences of categories.

We expect that this conjecture may be proved along the lines of the proofof Theorem 10.3.4 presented in Section 10.3.4. More precisely, it should followfrom the conjecture of [FG6], which generalizes the corresponding statementsin the case λ = 0.

Conjecture 10.3.13(1) Vλ is free as a zλ-module.(2)

Extibg,G[[t]](Vλ,Vλ) ' Ωizλ, (10.3.29)

the space of differential forms of degree i on OpλLG.

10.4 The tamely ramified case

In the previous section we have considered categorical analogues of the ir-reducible unramified representations of a reductive group G(F ) over a localnon-archimedian field F . We recall that these are the representations con-taining non-zero vectors fixed by the maximal compact subgroup K0 ⊂ G(F ).The corresponding Langlands parameters are unramified admissible homo-morphisms from the Weil–Deligne group W ′F to LG, i.e., those which factorthrough the quotient

W ′F →WF → Z,

and whose image in LG is semi-simple. Such homomorphisms are parameter-ized by semi-simple conjugacy classes in LG.

We have seen that the categorical analogues of unramified representationsof G(F ) are the categories gκc -modχ (equipped with an action of the loopgroup G((t))), where χ is a LG-oper on D× whose underlying LG-local systemis trivial. These categories can be called unramified in the sense that theycontain non-zero G[[t]]-equivariant objects. The corresponding Langlands pa-rameter is the trivial LG-local system σ0 on D×, which should be viewed asan analogue of an unramified homomorphism W ′F → LG. However, the local

10.4 The tamely ramified case 327

system σ0 is realized by many different opers, and this introduces an addi-tional complication into our picture: at the end of the day we need to showthat the categories gκc -modχ, where χ is of the above type, are equivalentto each other. In particular, Conjecture 10.3.10 describes what we expect tohappen when χ ∈ OpLG(D).

The next natural step is to consider categorical analogues of representationsof G(F ) that contain vectors invariant under the Iwahori subgroup I ⊂ G[[t]],the preimage of a fixed Borel subgroup B ⊂ G under the evaluation homomor-phism G[[t]]→ G. We begin this section by recalling a classification of theserepresentations, due to D. Kazhdan and G. Lusztig [KL] and V. Ginzburg[CG]. We then discuss the categorical analogues of these representations fol-lowing [FG2]–[FG5] and the intricate interplay between the classical and thegeometric pictures. We close this section with some explicit calculations inthe case of g = sl2, which should serve as an illustration of the general theory.

10.4.1 Tamely ramified representations

The Langlands parameters corresponding to irreducible representations ofG(F ) with I-invariant vectors are tamely ramified homomorphisms W ′F →LG. Recall from Section 1.1.3 that W ′F = WF n C. A homomorphismW ′F → LG is called tamely ramified if it factors through the quotient

W ′F → Z n C.

According to the relation (1.1.1), the group ZnC is generated by two elementsF = 1 ∈ Z (Frobenius) and M = 1 ∈ C (monodromy) satisfying the relation

FMF−1 = qM. (10.4.1)

Under an admissible tamely ramified homomorphism the generator F goesto a semi-simple element γ ∈ LG and the generator M goes to a unipotentelement N ∈ LG. According to formula (10.4.1), they have to satisfy therelation

γNγ−1 = Nq. (10.4.2)

Alternatively, we may write N = exp(u), where u is a nilpotent element ofLg. Then this relation becomes

γuγ−1 = qu.

Thus, we have the following bijection between the sets of equivalence classes

tamely ramified admissiblehomomorphisms W ′F → LG

⇐⇒ pairs γ ∈ LG, semi-simple,u ∈ Lg, nilpotent, γuγ−1 = qu

(10.4.3)


In both cases equivalence relation amounts to conjugation by an element ofLG.

Now to each Langlands parameter of this type we wish to attach an irre-ducible smooth representation of G(F ), which contains non-zero I-invariantvectors. It turns out that if G = GLn there is indeed a bijection, proved in[BZ], between the sets of equivalence classes of the following objects:

tamely ramified admissiblehomomorphisms W ′F → GLn

⇐⇒ irreducible representations(R, π) of GLn(F ), Rπ(I) 6= 0

(10.4.4)However, such a bijection is no longer true for other reductive groups: two

new phenomena appear, which we discuss presently.The first one is the appearance of L-packets. One no longer expects to be

able to assign to a particular admissible homomorphism W ′F → LG a singleirreducible smooth representation of G(F ). Instead, a finite collection of suchrepresentations (more precisely, a collection of equivalence classes of represen-tations) is assigned, called an L-packet. In order to distinguish representationsin a given L-packet, one needs to introduce an additional parameter. We willsee how this is done in the case at hand shortly. However, and this is the secondsubtlety alluded to above, it turns out that not all irreducible representationsof G(F ) within the L-packet associated to a given tamely ramified homomor-phism W ′F → LG contain non-zero I-invariant vectors. Fortunately, there isa certain property of the extra parameter used to distinguish representationsinside the L-packet that tells us whether the corresponding representation ofG(F ) has I-invariant vectors.

In the case of tamely ramified homomorphisms W ′F → LG this extra param-eter is an irreducible representation ρ of the finite group C(γ, u) of componentsof the simultaneous centralizer of γ and u in LG, on which the center of LG

acts trivially (see [Lu1]). In the case of G = GLn these centralizers are al-ways connected, and so this parameter never appears. But for other reductivegroups G this group of components is often non-trivial. The simplest exampleis when LG = G2 and u is a subprincipal nilpotent element of the Lie algebraLg.† In this case for some γ satisfying γuγ−1 = qu the group of componentsC(γ, u) is the symmetric group S3, which has three irreducible representations(up to equivalence). Each of them corresponds to a particular member of theL-packet associated with the tamely ramified homomorphism W ′F → LG de-fined by (γ, u). Thus, the L-packet consists of three (equivalence classes of)irreducible smooth representations of G(F ). However, not all of them containnon-zero I-invariant vectors.

The representations ρ of the finite group C(γ, u) which correspond to repre-sentations of G(F ) with I-invariant vectors are distinguished by the following

† The term “subprincipal” means that the adjoint orbit of this element has codimension 2in the nilpotent cone.


property. Consider the Springer fiber Spu. We recall (see formula (9.3.10))that

Spu = b′ ∈ LG/LB |u ∈ b′. (10.4.5)

The group C(γ, u) acts on the homology of the variety Spγu of γ-fixed points

of Spu. A representation ρ of C(γ, u) corresponds to a representation of G(F )with non-zero I-invariant vectors if and only if ρ occurs in the homology ofSpγ

u, H•(Spγu).

In the case of G2 the Springer fiber Spu of the subprincipal element u isa union of four projective lines connected with each other as in the Dynkindiagram of D4. For some γ the set Spγ

u is the union of a projective line(corresponding to the central vertex in the Dynkin diagram of D4) and threepoints (each in one of the remaining three projective lines). The correspondinggroup C(γ, u) = S3 on Spγ

u acts trivially on the projective line and by per-mutation of the three points. Therefore the trivial and the two-dimensionalrepresentations of S3 occur in H•(Spγ

u), but the sign representation does not.The irreducible representations of G(F ) corresponding to the first two con-tain non-zero I-invariant vectors, whereas the one corresponding to the signrepresentation of S3 does not.

The ultimate form of the local Langlands correspondence for representationsof G(F ) with I-invariant vectors is then as follows (here we assume, as in[KL, CG], that the group G is split and has connected center):

triples (γ, u, ρ), γuγ−1 = qu,

ρ ∈ Rep C(γ, u) occurs in H•(Spγu,C)

⇐⇒ irreducible representations(R, π) of G(F ), Rπ(I) 6= 0

(10.4.6)Again, this should be understood as a bijection between two sets of equivalenceclasses of the objects listed. This bijection is due to [KL] (see also [CG]). Itwas conjectured by Deligne and Langlands, with a subsequent modification(addition of ρ) made by Lusztig.

How to set up this bijection? The idea is to replace irreducible represen-tations of G(F ) appearing on the right hand side of (10.4.6) with irreduciblemodules over the corresponding Hecke algebra H(G(F ), I). Recall from Sec-tion 10.2.1 that this is the algebra of compactly supported I bi-invariantfunctions on G(F ), with respect to convolution. It naturally acts on thespace of I-invariant vectors of any smooth representation of G(F ) (see formula(10.2.2)). Thus, we obtain a functor from the category of smooth representa-tions of G(F ) to the category of H(G(F ), I). According to a theorem of A.Borel [Bor1], it induces a bijection between the set of equivalence classes ofirreducible smooth representations of G(F ) with non-zero I-invariant vectorsand the set of equivalence classes of irreducible H(G(F ), I)-modules.

The algebra H(G(F ), I) is known as the affine Hecke algebra and hasthe standard description in terms of generators and relations. However, for


our purposes we need another description, due to [KL, CG], which identifiesit with the equivariant K-theory of the Steinberg variety

St = N ×NN ,

where N ⊂ Lg is the nilpotent cone and N is the Springer resolution

N = x ∈ N , b′ ∈ LG/LB | x ∈ b′.

Thus, a point of St is a triple consisting of a nilpotent element of Lg and twoBorel subalgebras containing it. The group LG × C× naturally acts on St,with LG conjugating members of the triple and C× acting by multiplicationon the nilpotent elements,

a · (x, b′, b′′) = (a−1x, b′, b′′). (10.4.7)

According to a theorem of [KL, CG], there is an isomorphism

H(G(F ), I) ' KLG×C×(St). (10.4.8)

The right hand side is the LG×C×-equivariantK-theory of St. It is an algebrawith respect to a natural operation of convolution (see [CG] for details). It isalso a free module over its center, isomorphic to

KLG×C×(pt) = Rep LG⊗ C[q,q−1].

Under the isomorphism (10.4.8) the element q goes to the standard parameterq of the affine Hecke algebra H(G(F ), I) (here we consider H(G(F ), I) as aC[q,q−1]-module).

Now, the algebra KLG×C×(St), and hence the algebra H(G(F ), I), has a

natural family of modules, which are parameterized precisely by the conjugacyclasses of pairs (γ, u) as above. On these modules H(G(F ), I) acts via acentral character corresponding to a point in Spec(Rep LG⊗

CC[q,q−1]), which

is just a pair (γ, q), where γ is a semi-simple conjugacy class in LG andq ∈ C×. In our situation q is the cardinality of the residue field of F (hencea power of a prime), but in what follows we will allow a larger range ofpossible values of q: all non-zero complex numbers except for the roots ofunity. Consider the quotient of H(G(F ), I) by the central character definedby (γ, u). This is just the algebra K

LG×C×(St), specialized at (γ, q). Wedenote it by K

LG×C×(St)(γ,q).Now for a nilpotent element u ∈ N consider the Springer fiber Spu. The

condition that γuγ−1 = qu means that u, and hence Spu, is stabilized bythe action of (γ, q) ∈ LG × C× (see formula (10.4.7)). Let A be the smallestalgebraic subgroup of LG×C× containing (γ, q). The algebraK

LG×C×(St)(γ,q)

naturally acts on the equivariant K-theory KA(Spu) specialized at (γ, q),

KA(Spu)(γ,q) = KA(Spu) ⊗Rep A

C(γ,q).


It is known that KA(Spu)(γ,q) is isomorphic to the homology H•(Spγu) of the

γ-fixed subset of Spu (see [KL, CG]). Thus, we obtain that KA(Spu)(γ,q) is amodule over H(G(F ), I).

Unfortunately, theseH(G(F ), I)-modules are not irreducible in general, andone needs to work harder to describe the irreducible modules over H(G(F ), I).For G = GLn one can show that each of these modules has a unique irre-ducible quotient, and this way one recovers the bijection (10.4.4). But for ageneral groups G the finite groups C(γ, u) come into play. Namely, the groupC(γ, u) acts on KA(Spu)(γ,q), and this action commutes with the action ofK

LG×C×(St)(γ,q). Therefore we have a decomposition

KA(Spu)(γ,q) =⊕

ρ∈Irrep C(γ,u)

ρ⊗KA(Spu)(γ,q,ρ),

of KA(Spu)(γ,q) as a representation of C(γ, u) × H(G(F ), I). One shows(see [KL, CG] for details) that each H(G(F ), I)-module KA(Spu)(γ,q,ρ) has aunique irreducible quotient, and this way one obtains a parameterization ofirreducible modules by the triples appearing in the left hand side of (10.4.6).Therefore we obtain that the same set is in bijection with the right hand sideof (10.4.6). This is how the tame local Langlands correspondence (10.4.6),also known as the Deligne–Langlands conjecture, is proved.

10.4.2 Categories admitting (gκc , I) Harish-Chandra modules

We now wish to find categorical analogues of the above results in the frame-work of the categorical Langlands correspondence for loop groups.

As we explained in Section 10.2.2, in the categorical setting a representationof G(F ) is replaced by a category gκc -modχ equipped with an action of G((t)),and the space of I-invariant vectors is replaced by the subcategory of (gκc

, I)Harish-Chandra modules in gκc

-modχ. Hence the analogue of the question asto which representations of G(F ) admit non-zero I-invariant vectors becomesthe following question: for which χ does the category gκc -modχ contain non-zero (gκc

, I) Harish-Chandra modules? The answer is given by the followinglemma.

Recall that in Section 9.1.2 we introduced the space OpRSLG(D)$ of LG-opers

on D with regular singularity and residue $ ∈ Lh/W = h∗/W , where W isthe Weyl group of LG. Given µ ∈ h∗, we write $(µ) for the projection of µonto h∗/W . Finally, let P be the set of integral (not necessarily dominant)weights of g, viewed as a subset of h∗.

Lemma 10.4.1 The category gκc -modχ contains a non-zero (gκc , I) Harish-


Chandra module if and only if

χ ∈⊔

ν∈P/W

OpRSLG(D)$(ν). (10.4.9)

Proof. Let M be a non-zero (gκc , I) Harish-Chandra module in gκc -modχ.We show, in the same way as in the proof of Lemma 10.3.2, that it containsa vector v annihilated by I0 = [I, I] and such that I/I0 = H acts via acharacter ν ∈ P . Therefore there is a non-zero homomorphism from the Vermamodule Mν to M sending the highest weight vector of Mν to v. According toTheorem 9.5.3, the action of the center Z(g) ' FunOpLG(D×) on Mν factorsthrough FunOpRS

LG(D)$(−ν−ρ). Therefore χ ∈ OpRSLG(D)$(−ν−ρ).

On the other hand, suppose that χ belongs to OpRSLG(D)$(ν) for some ν ∈ P .

Let Mν(χ) be the quotient of the Verma module Mν by the central charactercorresponding to χ. It follows from Theorem 9.5.3 that Mν(χ) is a non-zeroobject of gκc -modχ. On the other hand, it is clear that Mν , and hence Mν(χ),are I-equivariant gκc

-modules.

Thus, the opers χ for which the corresponding category gκc -modχ containnon-trivial I-equivariant objects are precisely the points of the subscheme(10.4.9) of OpLG(D×). The next question is what are the corresponding LG-local systems.

Let LocRS,tameLG

⊂ LocLG(D×) be the locus of LG-local systems on D× withregular singularity and unipotent monodromy. Such a local system is deter-mined, up to an isomorphism, by the conjugacy class of its monodromy (see,e.g., [BV], Section 8). Therefore LocRS,tame

LGis an algebraic stack isomorphic

to N/LG. The following result is proved in a way similar to the proof ofLemma 10.3.1.

Lemma 10.4.2 If the local system underlying an oper χ ∈ OpLG(D×) belongsto LocRS,tame

LG, then χ belongs to the subset (10.4.9) of OpLG(D×).

In other words, the subscheme (10.4.9) is precisely the (set-theoretic) preim-age of LocRS,tame

LG⊂ LocLG(D×) under the map α : OpLG(D×)→ LocLG(D×).

This hardly comes as a surprise. Indeed, by analogy with the classical Lang-lands correspondence we expect that the categories gκc

-modχ containing non-trivial I-equivariant objects correspond to the Langlands parameters whichare the geometric counterparts of tamely ramified homomorphismsW ′F → LG.The most obvious candidates for those are precisely the LG-local systems onD× with regular singularity and unipotent monodromy. For this reason wewill call such local systems tamely ramified (this is reflected in the abovenotation).

Let us summarize: suppose that σ is a tamely ramified LG-local system onD×, and let χ be a LG-oper that is in the gauge equivalence class of σ. Then χbelongs to the subscheme (10.4.9), and the corresponding category gκc -modχ


contains non-zero I-equivariant objects, by Lemma 10.4.1. Let gκc -modIχ be

the corresponding category of I-equivariant (or, equivalently, (gκc, I) Harish-

Chandra) modules. This is our candidate for the categorification of the spaceof I-invariant vectors in an irreducible representation of G(F ) correspondingto a tamely ramified homomorphism W ′F → LG.

Note that according to Conjecture 10.1.2, the categories gκc -modχ (resp.,gκc

-modIχ) should be equivalent to each other for all χ which are gauge equiv-

alent to each other as LG-local systems.In the next section, following [FG2], we will give a conjectural description

of the categories gκc -modIχ for χ ∈ OpRS

LG(D)$(−ρ) in terms of the categoryof coherent sheaves on the Springer fiber corresponding to the residue of χ.This description in particular implies that at least the derived categories ofthese categories are equivalent to each other for the opers corresponding tothe same local system. We have a similar conjecture for χ ∈ OpRS

LG(D)$(ν)

for other ν ∈ P , which the reader may easily reconstruct from our discussionof the case ν = −ρ.

10.4.3 Conjectural description of the categories of (gκc , I)Harish-Chandra modules

Let us consider one of the connected components of the subscheme (10.4.9),namely, OpRS

LG(D)$(−ρ). Recall from Section 9.2.1 and Proposition 9.2.1 thatthis space is isomorphic to the space Opnilp

LGof nilpotent opers on D (cor-

responding to the weight λ = 0). As explained in Section 9.2.3, it comesequipped with the residue maps

ResF : OpnilpLG→ LnFLB,0

, Res : OpnilpLG→ Ln/LB = N/LG.

For any χ ∈ OpnilpLG

the LG-gauge equivalence class of the corresponding con-nection is a tamely ramified LG-local system on D×. Moreover, its mon-odromy conjugacy class is equal to exp(2πiRes(χ)).

We wish to describe the category gκc -modIχ of (gκc , I) Harish-Chandra

modules with the central character χ ∈ OpnilpLG

. However, here we face thefirst major complication as compared to the unramified case. While in theramified case we worked with the abelian category gκc -modG[[t]]

χ , this doesnot seem to be possible in the tamely ramified case. So from now on we willwork with the appropriate derived category Db(gκc -modχ)I . By definition,this is the full subcategory of the bounded derived category Db(gκc -modχ),whose objects are complexes with cohomologies in gκc

-modIχ.

Roughly speaking, the conjecture of [FG2] is that the categoryDb(gκc -modχ)I is equivalent to the derived category Db(QCoh(SpResF (χ)))of the category QCoh(SpResF (χ)) of quasicoherent sheaves on the Springerfiber of ResF (χ). However, we need to make some adjustments to this state-


ment. These adjustments are needed to arrive at a “nice” statement, Con-jecture 10.4.4 below. We now explain what these adjustments are and thereasons behind them.

The first adjustment is that we need to consider a slightly larger categoryof representations than Db(gκc -modχ)I . Namely, we wish to include exten-sions of I-equivariant gκc

-modules which are not necessarily I-equivariant,but only I0-equivariant, where I0 = [I, I]. To explain this more precisely,let us choose a Cartan subgroup H ⊂ B ⊂ I and the corresponding Lie sub-algebra h ⊂ b ⊂ Lie I. We then have an isomorphism I = H n I0. AnI-equivariant gκc

-module is the same as a module on which h acts diago-nally with eigenvalues given by integral weights and the Lie algebra Lie I0

acts locally nilpotently. However, there may exist extensions between suchmodules on which the action of h is no longer semi-simple. Such modulesare called I-monodromic. More precisely, an I-monodromic gκc

-module isa module that admits an increasing filtration whose consecutive quotients areI-equivariant. It is natural to include such modules in our category. However,it is easy to show that an I-monodromic object of gκc -modχ is the same as anI0-equivariant object of gκc

-modχ for any χ ∈ OpnilpLG

(see [FG2]). Thereforeinstead of I-monodromic modules we will use I0-equivariant modules. Denoteby Db(gκc -modχ)I0

the full subcategory of Db(gκc -modχ) whose objects arecomplexes with cohomologies in gκc -modI0

χ .The second adjustment has to do with the non-flatness of the Springer

resolution N → N . By definition, the Springer fiber Spu is the fiber productN ×N

pt, where pt is the point u ∈ N . This means that the structure sheaf of

Spu is given by

OSpu= O eN ⊗ON C. (10.4.10)

However, because the morphism N → N is not flat, this tensor productfunctor is not left exact, and there are non-trivial derived tensor products(the Tor’s). Our (conjectural) equivalence is not going to be an exact functor:it sends a general object of the category gκc -modI0

χ not to an object of thecategory of quasicoherent sheaves, but to a complex of sheaves, or, moreprecisely, an object of the corresponding derived category. Hence we are forcedto work with derived categories, and so the higher derived tensor productsneed to be taken into account.

To understand better the consequences of this non-exactness, let us considerthe following model example. Suppose that we have established an equivalencebetween the derived category Db(QCoh(N )) and another derived categoryDb(C). In particular, this means that both categories carry an action of thealgebra FunN (recall that N is an affine algebraic variety). Let us supposethat the action of FunN on Db(C) comes from its action on the abeliancategory C. Thus, C fibers over N , and let Cu the fiber category corresponding


to u ∈ N . This is the full subcategory of C whose objects are objects of Con which the ideal of u in FunN acts by 0.† What is the category Db(Cu)equivalent to?

It is tempting to say that it is equivalent to Db(QCoh(Spu)). However, thisdoes not follow from the equivalence of Db(QCoh(N )) and Db(C) becauseof the tensor product (10.4.10) having non-trivial higher derived functors.Suppose that the category C is flat over N ; this means that all projectiveobjects of C are flat as modules over FunN . Then Db(Cu) is equivalent to thecategory Db(QCoh(SpDG

u )), where SpDGu is the “DG fiber” of N → N at u.

By definition, a quasicoherent sheaf on SpDGu is a DG module over the DG

algebra

OSpDGu

= O eN L⊗ON

Cu, (10.4.11)

where we now take the full derived functor of tensor product. Thus, thecategoryDb(QCoh(SpDG

u )) may be thought of as the derived category of quasi-coherent sheaves on the “DG scheme” SpDG

u (see [CK] for a precise definitionof DG scheme).

Finally, the last adjustment is that we should consider the non-reducedSpringer fibers. This means that instead of the Springer resolution N weshould consider the “thickened” Springer resolution

Ñ = Lg ×LgN ,

where Lg is the so-called Grothendieck alteration,

Lg = x ∈ Lg, b′ ∈ LG/LB | x ∈ b′,

which we have encountered previously in Section 7.1.1. The variety Ñ is non-reduced, and the underlying reduced variety is the Springer resolution N .For instance, the fiber of N over a regular element in N consists of a single

point, but the corresponding fiber of Ñ is the spectrum of the Artinian ringh0 = Fun Lh/(Fun Lh)W

+ . Here (Fun Lh)W+ is the ideal in Fun Lh generated

by the augmentation ideal of the subalgebra of W -invariants. Thus, Spech0

is the scheme-theoretic fiber of $ : Lh → Lh/W at 0. It turns out that inorder to describe the categoryDb(gκc

-modχ)I0we need to use the “thickened”

Springer resolution.†Let us summarize: in order to construct the sought-after equivalence of

categories we take, instead of individual Springer fibers, the whole Springerresolution, and we further replace it by the “thickened” Springer resolution

† The relationship between C and Cu is similar to the relationship between bgκc -mod andand bgκc -modχ, where χ ∈ OpLG(D×).

† This is explained in Section 10.4.6 for g = sl2.

336 Constructing the Langlands correspondenceÑ defined above. In this version we will be able to formulate our equivalenceso that we avoid DG schemes.

This means that instead of considering the categories gκc-modχ for individ-

ual nilpotent opers χ, we should consider the “universal” category gκc-modnilp,

which is the “family version” of all of these categories. By definition, the cat-egory gκc -modnilp is the full subcategory of gκc -mod whose objects have theproperty that the action of Z(g) = FunOpLG(D) on them factors throughthe quotient Fun OpLG(D) → FunOpnilp

LG. Thus, the category gκc -modnilp is

similar to the category gκc-modreg that we have considered above. While the

former fibers over OpnilpLG

, the latter fibers over OpLG(D). The individual cat-egories gκc

-modχ are now realized as fibers of these categories over particularopers χ.

Our naive idea was that for each χ ∈ OpnilpLG

the category Db(gκc -modχ)I0

is equivalent to QCoh(SpResF (χ)). We would like to formulate now a “familyversion” of such an equivalence. To this end we form the fiber product

Lñ = Lg ×Lg

Ln.

It turns out that this fiber product does not suffer from the problem of theindividual Springer fibers, as the following lemma shows:

Lemma 10.4.3 ([FG2],Lemma 6.4) The derived tensor product

Fun LgL⊗

Fun LgFun Ln

is concentrated in cohomological dimension 0.

The variety Lñ may be thought of as the family of (non-reduced) Springerfibers parameterized by Ln ⊂ Lg. It is important to note that it is singular,reducible, and non-reduced. For example, if g = sl2, it has two components,one of which is P1 (the Springer fiber at 0) and the other is the doubled affineline (i.e., Spec C[x, y]/(y2)), see Section 10.4.6.

We note that the corresponding reduced scheme is Ln introduced in Sec-tion 9.3.2:

Ln = N ×N

Ln. (10.4.12)

However, the derived tensor product corresponding to (10.4.12) is not con-centrated in cohomological dimension 0, and this is the reason why we preferto use Lñ rather than Ln.

Now we set

MOp0LG = Opnilp

LG×

Ln/LB

Lñ/LB,

where we use the residue morphism Res : OpnilpLG→ Ln/LB. Thus, informally


MOp0LG may be thought as the family over Opnilp

LGwhose fiber over χ ∈ Opnilp

LG

is the Springer fiber of Res(χ).As the notation suggests, MOp0

LG is closely related to the space of Miuraopers whose underlying opers are nilpotent (see Section 10.4.5 below). Moreprecisely, the reduced scheme of MOp0

LG is the scheme MOp0LG of nilpotent

Miura LG-opers of weight 0, introduced in Section 9.3.2:

MOp0G ' Opnilp

LG×

Ln/LB

Ln/B

(see formulas (9.3.9) and (10.4.12)).We also introduce the category gκc -modI0

nilp as the full subcategory ofgκc

-modnilp whose objects are I0-equivariant, and the corresponding derivedcategory Db(gκc

-modnilp)I0.

Now we can formulate the Main Conjecture of [FG2]:

Conjecture 10.4.4 There is an equivalence of categories

Db(gκc -modnilp)I0' Db(QCoh(MOp0

LG)), (10.4.13)

which is compatible with the action of the algebra FunOpnilpLG

on both cate-gories.

Note that the action of FunOpnilpLG

on the first category comes from theaction of the center Z(g), and on the second category it comes from the factthat MOp0

LG is a scheme over OpnilpLG

.The conjectural equivalence (10.4.13) may be viewed as a categorical ana-

logue of the local Langlands correspondence in the tamely ramified case, aswe discuss in the next section.

An important feature of the equivalence (10.4.13) does not preserve the t-structure on the two categories. In other words, objects of the abelian categorygκc -modI0

nilp are in general mapped under this equivalence to complexes in

Db(QCoh(MOp0LG)), and vice versa. We will see examples of this in the case

of sl2 in Section 10.4.6.There are similar conjectures for the categories corresponding to the spaces

Opnilp,λLG

of nilpotent opers with dominant integral weights λ ∈ P+.In the next section we will discuss the connection between Conjecture 10.4.4

and the classical tamely ramified Langlands correspondence. We then presentsome evidence for this conjecture and consider in detail the example of g = sl2.

10.4.4 Connection between the classical and the geometric settings

Let us discuss the connection between the equivalence (10.4.13) and the re-alization of representations of affine Hecke algebras in terms of K-theory ofthe Springer fibers. As we have explained, we would like to view the cat-egory Db(gκc -modχ)I0

for χ ∈ OpnilpLG

as, roughly, a categorification of the


space Rπ(I) of I-invariant vectors in an irreducible representation (R, π) ofG(F ). Therefore, we expect that the Grothendieck group of the categoryDb(gκc -modχ)I0

is somehow related to the space Rπ(I).Let us try to specialize the statement of Conjecture 10.4.4 to a particular

oper χ = (F ,∇,FLB) ∈ OpnilpLG

. Let SpDG

ResF (χ) be the DG fiber of MOp0LG

over χ. By definition (see Section 9.2.3), the residue ResF (χ) of χ is a vector

in the twist of Ln by the LB-torsor FLB,0. It follows that SpDG

ResF (χ) is the DGfiber over ResF (χ) of the FLB,0-twist of the Grothendieck alteration.

If we trivialize FLB,0, then u = ResF (χ) becomes an element of Ln. By

definition, the (non-reduced) DG Springer fiber SpDG

u is the DG fiber of theGrothendieck alteration Lg → Lg at u. In other words, the correspondingstructure sheaf is the DG algebra

OfSpDGu

= OLeg L⊗OLg

Cu

(compare with formula (10.4.11)).To see what these DG fibers look like, let u = 0. Then the naive Springer

fiber is just the flag variety LG/LB (it is reduced in this case), and OfSp0

is the structure sheaf of LG/LB. But the sheaf OfSpDG0

is a sheaf of DGalgebras, which is quasi-isomorphic to the complex of differential forms onLG/LB, with the zero differential. In other words, Sp

DG

0 may be viewed as a“Z-graded manifold” such that the corresponding supermanifold, obtained byreplacing the Z-grading by the corresponding Z/2Z-grading, is ΠT (LG/LB),the tangent bundle to LG/LB with the parity of the fibers changed from evento odd.

We expect that the category gκc -modI0

nilp is flat over OpnilpLG

. Therefore,specializing Conjecture 10.4.4 to a particular oper χ ∈ Opnilp

LG, we obtain as a

corollary an equivalence of categories

Db(gκc -modχ)I0' Db(QCoh(Sp

DG

ResF (χ))). (10.4.14)

This fits well with Conjecture 10.1.2 saying that the categories gκc -modχ1

and gκc-modχ2 (and hence Db(gκc -modχ1)

I0and Db(gκc -modχ2)

I0) should

be equivalent if the underlying local systems of the opers χ1 and χ2 are iso-morphic. For nilpotent opers χ1 and χ2 this is so if and only if their mon-odromies are conjugate to each other. Since their monodromies are obtainedby exponentiating their residues, this is equivalent to saying that the residues,ResF (χ1) and ResF (χ2), are conjugate with respect to the FLB,0-twist of LG.But in this case the DG Springer fibers corresponding to χ1 and χ2 are alsoisomorphic, and so Db(gκc

-modχ1)I0

and Db(gκc -modχ2)I0

are equivalent toeach other, by (10.4.14).

The Grothendieck group of the category Db(QCoh(SpDG

u )), where u is any


nilpotent element, is the same as the Grothendieck group of QCoh(Spu). Inother words, the Grothendieck group does not “know” about the DG or the

non-reduced structure of SpDG

u . Hence it is nothing but the algebraic K-theory K(Spu). As we explained at the end of Section 10.4.1, equivariantvariants of this algebraic K-theory realize the “standard modules” over theaffine Hecke algebra H(G(F ), I). Moreover, the spaces of I-invariant vectorsRπ(I) as above, which are naturally modules over the affine Hecke algebra, maybe realized as subquotients of K(Spu). This indicates that the equivalences(10.4.14) and (10.4.13) are compatible with the classical results.

However, at first glance there are some important differences between theclassical and the categorical pictures, which we now discuss in more detail.

In the construction ofH(G(F ), I)-modules outlined in Section 10.4.1 we hadto pick a semi-simple element γ of LG such that γuγ−1 = qu, where q is thenumber of elements in the residue field of F . Then we consider the specializedA-equivariant K-theory KA(Spu)(γ,q), where A is the the smallest algebraicsubgroup of LG × C× containing (γ, q). This gives K(Spu) the structure ofan H(G(F ), I)-module. But this module carries a residual symmetry withrespect to the group C(γ, u) of components of the centralizer of γ and u inLG, which commutes with the action of H(G(F ), I). Hence we consider theH(G(F ), I)-module

KA(Spu)(γ,q,ρ) = HomC(γ,u)(ρ,K(Spu)),

corresponding to an irreducible representation ρ of C(γ, u). Finally, each ofthese components has a unique irreducible quotient, and this is an irreduciblerepresentation of H(G(F ), I), which is realized on the space Rπ(I), where(R, π) is an irreducible representation of G(F ) corresponding to (γ, u, ρ) un-der the bijection (10.4.6). How is this intricate structure reflected in thecategorical setting?

Our category Db(QCoh(SpDG

u )), where u = ResF (χ), is a particular cat-egorification of the (non-equivariant) K-theory K(Spu). Note that in theclassical local Langlands correspondence (10.4.6), the element u of the triple(γ, u, ρ) is interpreted as the logarithm of the monodromy of the correspondingrepresentation of the Weil–Deligne group W ′F . This is in agreement with theinterpretation of ResF (χ) as the logarithm of the monodromy of the LG-localsystem on D× corresponding to χ, which plays the role of the local Langlandsparameter for the category gκc -modχ (up to the inessential factor 2πi).

But what about the other parameters, γ and ρ? And why does our categorycorrespond to the non-equivariant K-theory of the Springer fiber, and not theequivariant K-theory, as in the classical setting?

The element γ corresponding to the Frobenius in W ′F does not seem to havean analogue in the geometric setting. We have already seen this above in theunramified case: while in the classical setting unramified local Langlands


parameters are the semi-simple conjugacy classes γ in LG, in the geometricsetting we have only one unramified local Langlands parameter; namely, thetrivial local system.

To understand better what is going on here, we revisit the unramified case.Recall that the spherical Hecke algebra H(G(F ),K0) is isomorphic to the rep-resentation ring Rep LG. The one-dimensional space of K0-invariants in an ir-reducible unramified representation (R, π) of G(F ) realizes a one-dimensionalrepresentation of H(G(F ),K0), i.e., a homomorphism Rep LG→ C. The un-ramified Langlands parameter γ of (R, π), which is a semi-simple conjugacyclass in LG, is the point in Spec(Rep LG) corresponding to this homomor-phism. What is a categorical analogue of this homomorphism? The cate-gorification of Rep LG is the category Rep LG. The product structure onRep LG is reflected in the structure of tensor category on Rep LG. On theother hand, the categorification of the algebra C is the category Vect of vectorspaces. Therefore a categorical analogue of a homomorphism Rep LG→ C isa functor Rep LG→ Vect respecting the tensor structures on both categories.Such functors are called the fiber functors. The fiber functors form a categoryof their own, which is equivalent to the category of LG-torsors. Thus, any twofiber functors are isomorphic, but not canonically. In particular, the group ofautomorphisms of each fiber functor is isomorphic to LG. (Incidentally, thisis how LG is reconstructed from a fiber functor in the Tannakian formalism.)Thus, we see that, while in the categorical world we do not have analogues ofsemi-simple conjugacy classes γ (the points of Spec(Rep LG)), their role is insome sense played by the group of automorphisms of a fiber functor.

This is reflected in the fact that, while in the categorical setting we havea unique unramified Langlands parameter – namely, the trivial LG-local sys-tem σ0 on D× – this local system has a non-trivial group of automorphisms,namely, LG. We therefore expect that the group LG should act by automor-phisms of the Langlands category Cσ0 corresponding to σ0, and this actionshould commute with the action of the loop group G((t)) on Cσ0 . It is thisaction of LG that is meant to compensate for the lack of unramified Langlandsparameters, as compared to the classical setting.

We have argued in Section 10.3 that the category gκc -modχ, where χ =(F ,∇,FLB) ∈ OpLG(D), is a candidate for the Langlands category Cσ0 .Therefore we expect that the group LG (more precisely, its twist LGF ) actson the category gκc

-modχ. In Section 10.3.7 we showed how to obtain thisaction using the conjectural equivalence between gκc

-modχ and the categoryDHecke

κc-modχ of Hecke eigenmodules on the affine Grassmannian Gr (see Con-

jecture 10.3.9). The category DHeckeκc

-modχ was defined in Section 10.3.6 asa “de-equivariantization” of the category Dκc -mod of twisted D-modules onGr with respect to the monoidal action of the category Rep LG.

Now comes a crucial observation that will be useful for understanding theway things work in the tamely ramified case: the category Rep LG may be


interpreted as the category of LG-equivariant quasicoherent sheaves on thevariety pt = Spec C. In other words, Rep LG may be interpreted as the cate-gory of quasicoherent sheaves on the stack pt /LG. The existence of monoidalaction of the category Rep LG on Dκc -mod should be viewed as the statementthat the category Dκc

-mod “fibers” over the stack pt /LG. The statement ofConjecture 10.3.9 may then be interpreted as saying that

gκc-modχ ' Dκc

-mod ×pt /LG

pt .

In other words, if C is the conjectural Langlands category fibering over thestack LocLG(D×) of all LG-local systems on D×, then

Dκc -mod ' C ×LocLG(D×)

pt /LG,

whereas

gκc -modχ ' C ×LocLG(D×)

pt,

where the morphism pt→ LocLG(D×) corresponds to the oper χ.Thus, in the categorical setting there are two different ways to think about

the trivial local system σ0: as a point (defined by a particular LG-bundleon D with connection, such as a regular oper χ), or as a stack pt /LG. Thebase change of the Langlands category in the first case gives us a categorywith an action of LG, such as the categories gκc -modχ or DHecke

κc-mod. The

base change in the second case gives us a category with a monoidal action ofRep LG, such as the category Dκc -mod. We can go back and forth betweenthe two versions by applying the procedures of equivariantization and de-equivariantization with respect to LG and Rep LG, respectively.

Now we return to the tamely ramified case. The semi-simple element γ ap-pearing in the triple (γ, u, ρ) plays the same role as the unramified Langlandsparameter γ. However, now it must satisfy the identity γuγ−1 = qu. Recallthat the center Z of H(G(F ), I) is isomorphic to Rep LG, and so SpecZ isthe set of all semi-simple elements in LG. For a fixed nilpotent element u theequation γuγ−1 = qu cuts out a locus Cu in SpecZ corresponding to thosecentral characters which may occur on irreducible H(G(F ), I)-modules corre-sponding to u. In the categorical setting (where we set q = 1) the analogue ofCu is the centralizer Z(u) of u in LG, which is precisely the group Aut(σ) ofautomorphisms of a tame local system σ on D× with monodromy exp(2πiu).On general grounds we expect that the group Aut(σ) acts on the Langlandscategory Cσ, just as we expect the group LG of automorphisms of the triviallocal system σ0 to act on the category Cσ0 . It is this action that replaces theparameter γ in the geometric setting.

In the classical setting we also have one more parameter, ρ. Let us recallthat ρ is a representation of the group C(γ, u) of connected components of the


centralizer Z(γ, u) of γ and u. But the group Z(γ, u) is a subgroup of Z(u),which becomes the group Aut(σ) in the geometric setting. Therefore one canargue that the parameter ρ is also absorbed into the action of Aut(σ) on thecategory Cσ.

If we have an action of Aut(σ) on the category Cσ, or on one of its manyincarnations gκc

-modχ, χ ∈ OpnilpLG

, it means that these categories must be“de-equivariantized,” just like the categories gκc

-modχ, χ ∈ OpLG(D), in theunramified case. This is the reason why in the equivalence (10.4.14) (and inConjecture 10.4.4) we have the non-equivariant categories of quasicoherentsheaves (whose Grothendieck groups correspond to the non-equivariant K-theory of the Springer fibers).

However, there is also an equivariant version of these categories. Considerthe substack LocRS,tame

LGof tamely ramified local systems in LocLG(D×) in-

troduced in Section 10.4.2. Since a tamely ramified local system is completelydetermined by the logarithm of its (unipotent) monodromy, this substack isisomorphic to N/LG. This substack plays the role of the substack pt /LG

corresponding to the trivial local system. Let us set

Ctame = C ×LocLG(D×)

N/LG.

Then, according to our general conjecture expressed by the Cartesian diagram(10.1.3), we expect to have

gκc -modnilp ' Ctame ×N/LG

OpnilpLG

. (10.4.15)

LetDb(Ctame)I0be the I0-equivariant derived category corresponding to Ctame.

Combining (10.4.15) with Conjecture 10.4.4, and noting that

MOp0LG ' Opnilp

LG×N/LG

Ñ/LG,

we obtain the following conjecture (see [FG2]):

Db(Ctame)I0' Db(QCoh( Ñ/LG)). (10.4.16)

The category on the right hand side may be interpreted as the derived cat-egory of LG-equivariant quasicoherent sheaves on the “thickened” Springer

resolution Ñ .Together, the conjectural equivalences (10.4.14) and (10.4.16) should be

thought of as the categorical versions of the realizations of modules over theaffine Hecke algebra in the K-theory of the Springer fibers.

One corollary of the equivalence (10.4.14) is the following: the classes ofirreducible objects of the category gκc -modI0

χ in the Grothendieck groupof gκc -modI0

χ give rise to a basis in the algebraic K-theory K(Spu), whereu = ResF (χ). Presumably, this basis is closely related to the bases in (the


equivariant version of) this K-theory constructed by G. Lusztig in [Lu2] (fromthe perspective of unrestricted g-modules in positive characteristic).

10.4.5 Evidence for the conjecture

We now describe some evidence for Conjecture 10.4.4. It consists of the fol-lowing four groups of results:

• Interpretation of the Wakimoto modules as gκc -modules corresponding tothe skyscraper sheaves on MOp0

LG.• Proof of the equivalence of certain quotient categories of Db(gκc -modnilp)I0

and Db(QCoh(MOp0LG)).

• Proof of the restriction of the equivalence (10.4.13) to regular opers.• Connection to R. Bezrukavnikov’s theory.

We start with the discussion of Wakimoto modules.Suppose that we have proved the equivalence of categories (10.4.13). Then

each quasicoherent sheaf on MOp0LG should correspond to an object of the

derived category Db(gκc -modnilp)I0. The simplest quasicoherent sheaves on

MOp0LG are the skyscraper sheaves supported at the C-points of MOp0

LG.It follows from the definition that a C-point of MOp0

LG, which is the same as aC-point of the reduced scheme MOp0

G, is a pair (χ, b′), where χ = (F ,∇,FLB)is a nilpotent LG-oper in Opnilp

LGand b′ is a point of the Springer fiber corre-

sponding to ResF (χ), which is the variety of Borel subalgebras in LgF0 thatcontain ResF (χ). Thus, if Conjecture 10.4.4 is true, we should have a familyof objects of the category Db(gκc

-modnilp)I0parameterized by these data.

What are these objects?The reader who has studied the previous chapter of this book will no

doubt have already guessed the answer: these are the Wakimoto mod-ules! Indeed, let us recall that Wakimoto modules of critical level are pa-rameterized by the space Conn(Ω−ρ)D× . The Wakimoto module correspond-ing to ∇ ∈ Conn(Ω−ρ)D× is denoted by W∇ (see Section 9.4.3). Accord-ing to Theorem 8.3.3, the center Z(g) acts on W∇ via the central characterµ(∇), where µ is the Miura transformation. Moreover, as explained in Sec-tion 9.4.3, if χ ∈ Opnilp

LG, then W∇ is an object of the category gκc -modI

χ forany ∇ ∈ µ−1(χ).

According to Theorem 9.3.7, the points of µ−1(χ) are in bijection with thepoints of the Springer fiber SpResF (χ) corresponding to the nilpotent elementResF (χ). Therefore to each point of SpResF (χ) we have assigned a Wakimoto

module, which is an object of the category gκc -modI0

χ (and hence of the cor-responding derived category). In other words, Wakimoto modules are objectsof the category gκc

-modInilp parameterized by the C-points of MOp0

LG. It is

natural to assume that they correspond to the skyscraper sheaves on MOp0LG


under the equivalence (10.4.13). This was in fact one of our motivations forthis conjecture.

Incidentally, this gives us a glimpse into how the group of automorphismsof the LG-local system underlying the oper χ acts on the category gκc

-modχ.This group is Z(ResF (χ)), the centralizer of the residue ResF (χ), and itacts on the Springer fiber SpResF (χ). Therefore g ∈ Z(ResF (χ)) sends theskyscraper sheaf supported at a point p ∈ SpResF (χ) to the skyscraper sheafsupported at g · p. Hence we expect that g sends the Wakimoto modulecorresponding to p to the Wakimoto module corresponding to g · p.

If the Wakimoto modules indeed correspond to the skyscraper sheaves, thenthe equivalence (10.4.13) may be thought of as a kind of “spectral decompo-sition” of the category Db(gκc

-modnilp)I0, with the basic objects being the

Wakimoto modules W∇, where ∇ runs over the locus in Conn(Ω−ρ)D× , whichis isomorphic, pointwise, to MOp0

LG.Note however that, even though the (set-theoretic) preimage of Opnilp

LGin

Conn(Ω−ρ)D× under the Miura transformation µ is equal to MOp0G, they

are not isomorphic as algebraic varieties. Instead, according to Section 9.3.3,µ−1(Opnilp

LG) is the disjoint union of subvarieties

µ−1(OpnilpLG

) =⊔

w∈W

Conn(Ω−ρ)w(ρ)−ρD ,

where Conn(Ω−ρ)w(ρ)−ρD is the variety of connections with regular singularity

and residue w(ρ)−ρ. On the other hand, MOp0LG has a stratification (obtained

from intersections with the Schubert cells on the flag variety LG/LB), withthe strata MOp0,w

LGisomorphic to Conn(Ω−ρ)w(ρ)−ρ

D , w ∈W (see Section 9.3.2and Theorem 9.3.6). But these strata are “glued” together in MOp0

LG in anon-trivial way.

For each stratum, which is an affine algebraic variety (in fact, an infinite-dimensional affine space), we can associate a functor

Fw : QCoh(Conn(Ω−ρ)w(ρ)−ρD )→ gκc -modI0

nilp,

using the Wakimoto module Ww(ρ)−ρ defined by formula (9.4.3). The functorFw maps an object of QCoh(Conn(Ω−ρ)w(ρ)−ρ

D ), which is the same as a moduleF over the algebra FunConn(Ω−ρ)w(ρ)−ρ

D , to the gκc -module

Ww(ρ)−ρ ⊗Fun Conn(Ω−ρ)

w(ρ)−ρD

F .

Informally, one can say that the category Db(QCoh(MOp0LG)) is “glued”

from the categories of quasi-coherent sheaves on the strata Conn(Ω−ρ)w(ρ)−ρD .

Conjecture 10.4.4 may therefore be interpreted as saying that the categoryDb(gκc -modnilp)I0

is “glued” from its subcategories obtained as the imagesof these functors. For more on this, see [FG5].


At this point it is instructive to compare the conjectural equivalence (10.4.13)with the equivalence (10.3.9) in the case of regular opers, which we rewrite ina more suggestive way as

gκc-modG[[t]]

reg ' QCoh(OpLG(D)). (10.4.17)

There are two essential differences between the two statements. First of all,(10.4.17) is an equivalence between abelian categories, whereas in (10.4.13)we need to use the derived categories. The second difference is that while in(10.4.17) we have the category of quasicoherent sheaves on a smooth affinealgebraic variety OpLG(D) (actually, an infinite-dimensional affine space), in(10.4.13) we have the category of quasicoherent sheaves on a quasi-projectivevariety MOp0

LG which is highly singular and non-reduced. In some sense,most of the difficulties in proving (10.4.13) are caused by the non-affinenessof MOp0

LG.However, there is a truncated version of the equivalence (10.4.13), which in-

volves the category of quasicoherent sheaves on an affine subscheme of MOp0LG

and a certain quotient of the categoryDb(gκc -modnilp)I0. Let us give a precise

statement.For each i = 1, . . . , `, let sl

(i)2 be the sl2 subalgebra of g = g⊗ 1 ⊂ gκc gen-

erated by ei, hi, fi. Let us call a gκc -module M partially integrable if thereexists i = 1, . . . , `, such that the action of sl

(i)2 on M may be exponentiated

to an action of the corresponding group SL2. It easy to see that partiallyintegrable objects form a Serre subcategory in gI0

nilp. Let f gκc -modI0

nilp be the

quotient category of gκc -modI0

nilp by the subcategory of partially integrableobjects. We will denote by fDb(gκc -modnilp)I0

the triangulated quotient cat-egory of Db(gκc -modnilp)I0

by the subcategory whose objects have partiallyintegrable cohomology.

On the other hand, consider the subscheme MOp0,w0LG

of MOp0LG, which is

the closure of the restriction of MOp0LG to the locus of regular nilpotent ele-

ments in Ln under the map MOp0LG →

Ln. As we have seen in Section 10.4.3,the non-reduced Springer fiber over regular nilpotent elements is isomorphic toSpech0, where h0 = Fun Lh/(Fun Lh)W

+ is the Artinian algebra of dimension|W |. Therefore we have

MOp0,w0LG' Opnilp

LG×Spech0.

Thus, we see that it is an affine subscheme of MOp0LG. The corresponding

reduced scheme is in fact the stratum MOp0,w0LG⊂ MOp0

LG introduced in Sec-tion 9.3.2. The category of quasicoherent sheaves on MOp0,w0

LGmay be realized

as a Serre quotient of the category of quasicoherent sheaves on MOp0LG.

The following result, proved in [FG2], shows that the two quotient categoriesare equivalent to each other (already at the level of abelian categories).


Theorem 10.4.5 The categories f gκc -modI0

nilp and QCoh(MOp0,w0LG

) are equiv-alent to each other.

This is the second piece of evidence for Conjecture 10.4.4 from the list givenat the beginning of this section.

Let us now discuss the third piece of evidence. The categories on both sidesof (10.4.13) fiber over Opnilp

LG; in other words, the algebra FunOpnilp

LGacts on

objects of each of these categories. Consider the restrictions of these categoriesto the subscheme of regular opers OpLG(D) ⊂ Opnilp

LG. The corresponding

abelian categories are defined as the full subcategories on which the action ofFunOpnilp

LGfactors through FunOpLG(D). The objects of the corresponding

derived categories are those complexes whose cohomologies have this property.The restriction of the category on the right hand side of (10.4.13) to the

subscheme OpLG(D) is the derived category of quasicoherent sheaves on theDG scheme

MOpregLG = MOp0

LG

L×

OpnilpLG

OpLG(D).

The corresponding tensor product functor is not left exact, and so we are inthe situation described in Section 10.4.3. Therefore we need to take the DGfiber product. Since the residue of any regular oper is equal to 0, we find thatthe fiber of MOp

regLG over each oper χ ∈ OpLG(D) is isomorphic to the DG

Springer fiber SpDG0 , which is the DG scheme ΠT (LG/LB), as we discussed

in Section 10.4.3.On the other hand, the restriction of the category on the left hand side of

(10.4.13) to OpLG(D) is the category Db(gκc -modreg)I0, the I0-equivariant

part of the derived category of gκc -modreg. According to a theorem provedin [FG4] (which is the I0-equivariant version of Conjecture 10.3.9), there isan equivalence of categories

gκc -modI0

reg ' DHeckeκc

-modI0

reg

and hence the corresponding derived categories are also equivalent. On theother hand, it follows from the results of [ABG] that the derived category ofDb(DHecke

κc-modreg)I0

is equivalent to Db(QCoh(MOpregLG)). Therefore the re-

strictions of the categories appearing in (10.4.13) to OpLG(D) are equivalent.Finally, we discuss the fourth piece of evidence, connection with Bezrukav-

nikov’s theory.To motivate it, let us recall that in Section 10.3.5 we discussed the action

of the categorical spherical Hecke algebra H(G((t)), G[[t]]) on the categorygκc -modχ, where χ is a regular oper. The affine Hecke algebraH(G(F ), I) alsohas a categorical analogue. Consider the affine flag variety Fl = G((t))/I.The categorical affine Hecke algebra is the category H(G((t)), I), which isthe full subcategory of the derived category of D-modules on Fl = G((t))/I


whose objects are complexes with I-equivariant cohomologies. This categorynaturally acts on the derived category Db(gκc

-modχ)I . What does this actioncorrespond to on the other side of the equivalence (10.4.13)?

The answer is given by a theorem of R. Bezrukavnikov [B], which may beviewed as a categorification of the isomorphism (10.4.8):

Db(DFlκc

-mod)I0' Db(QCoh(St)), (10.4.18)

where DFlκc

-mod is the category of twisted D-modules on Fl and St is the“thickened” Steinberg variety

St = N ×N

Ñ = N ×Lg

Lg.

Morally, we expect that the two categories in (10.4.18) act on the twocategories in (10.4.13) in a compatible way. However, strictly speaking, theleft hand side of (10.4.18) acts like this:

Db(gκc -modnilp)I → Db(gκc -modnilp)I0,

and the right hand side of (10.4.18) acts like this:

Db(QCoh(MOp0LG))→ Db(QCoh(MOp0

LG)).

So one needs a more precise statement, which may be found in [B], Section4.2. Alternatively, one can consider the corresponding actions of the affinebraid group of LG, as in [B].

A special case of this compatibility concerns some special objects of thecategory Db(DFl

κc-mod)I , the central sheaves introduced in [Ga1]. They cor-

respond to the central elements of the affine Hecke algebra H(G(F ), I). Thesecentral elements act as scalars on irreducible H(G(F ), I)-modules, as well ason the standard modules KA(Spu)(γ,q,ρ) discussed above. We have argued

that the categories gκc -modI0

χ , χ ∈ OpnilpLG

, are categorical versions of theserepresentations. Therefore it is natural to expect that its objects are “eigen-modules” with respect to the action of the central sheaves fromDb(DFl

κc-mod)I

(in the sense of Section 10.3.5). This has indeed been proved in [FG3].This discussion indicates the intimate connection between the categories

Db(gκc-modnilp) and the category of twisted D-modules on the affine flag

variety, which is similar to the connection between gκc-modreg and the cat-

egory of twisted D-modules on the affine Grassmannian, which we discussedin Section 10.3.6. A more precise conjecture relating Db(gκc -modnilp) andDb(DFl

κc-mod) was formulated in [FG2] (see the Introduction and Section 6),

where we refer the reader for more details. This conjecture may be viewedas an analogue of Conjecture 10.3.9 for nilpotent opers. As explained in[FG2], this conjecture is supported by the results of [AB, ABG] (see also [B]).Together, these results and conjectures provide additional evidence for theequivalence (10.4.13).


10.4.6 The case of g = sl2

In this section we will illustrate Conjecture 10.4.4 in the case when g = sl2,thus explaining the key ingredients of this conjecture in more concrete terms.

First, let us discuss the structure of the scheme MOp0PGL2

. By definition,

MOp0PGL2

= OpnilpPGL2

×n/PGL2

ñ/PGL2.

The variety ñ maps to n = C, and its fiber over u ∈ n is the non-reducedSpringer fiber of u, the variety of Borel subalgebras of sl2 containing u.

There are two possibilities. If u = 0, then the fiber is P1, the flag vari-ety of PGL2. If u 6= 0, then it corresponds to a regular nilpotent element,and therefore there is a unique Borel subalgebra containing u. Therefore thereduced Springer fiber is a single point. However, the non-reduced Springerfiber is a double point Spec C[ε]/(ε2) = Spech0.

To see this, let us show that there are non-trivial C[ε]/(ε2)-points of theGrothendieck alteration sl2 which project onto a regular nilpotent element insl2. Without loss of generality we may choose this element to be representedby the matrix

e =(

0 10 0

).

Then a C[ε]/(ε2)-point of sl2 above e is a Borel subalgebra in sl2[ε]/(ε2) whosereduction modulo ε contains e. Now observe that the Lie algebra b[ε]/(ε2) ofupper triangular matrices in sl2[ε]/(ε2) contains the element(

1 0−aε 1

)e

(1 0aε 1

)=(aε 10 −aε

)for any a ∈ C. Therefore

e ∈(

1 0−aε 1

)b[ε]/(ε2)

(1 0aε 1

), a ∈ C.

Hence we find non-trivial C[ε]/(ε2)-points of the Springer fiber over e.Thus, we obtain that the variety ñ has two components: one of them is P1

and the other one is a double affine line A1. These components correspond totwo elements of the Weyl group of sl2, which label possible relative positionsof two Borel subalgebras in sl2. The points on the line correspond to thecase when the Borel subalgebra coincides with the fixed Borel subalgebra b

containing n, and generic points of P1 correspond to Borel subalgebras thatare in generic relative position with b. These two components are “gluedtogether” at one point. Thus, we find that ñ is a singular reducible non-reduced quasi-projective variety.

This gives us a rough idea as to what MOp0PGL2

looks like. It fibers over the


space OpnilpPGL2

of nilpotent PGL2-opers. Let us choose a coordinate t on thedisc and use it to identify Opnilp

PGL2with the space of second order operators

∂2t − v(t), v(t) =

∑n≥−1

vntn.

It is easy to see that the residue map OpnilpPGL2

→ n = C is given by theformula v(t) 7→ v−1. Therefore, if we decompose Opnilp

PGL2into the Cartesian

product of a line with the coordinate v−1 and the infinite-dimensional affinespace with the coordinates vn, n ≥ 0, we obtain that MOp0

PGL2is isomorphic

to the Cartesian product of ñ and an infinite-dimensional affine space.Thus, MOp0

PGL2has two components, MOp0,1

PGL2and MOp0,w0

PGL2corre-

sponding to two elements, 1 and w0, of the Weyl group of sl2.Conjecture 10.4.4 states that the derived category of quasicoherent sheaves

on the scheme MOp0PGL2

is equivalent to the equivariant derived categoryDb(sl2 -modnilp)I0

. We now consider examples of objects of these categories.First, we consider the skyscraper sheaf at a point of MOp0

PGL2, which is

the same as a point of MOp0PGL2

. The variety MOp0PGL2

is a union of twostrata

MOp0PGL2

= MOp0,1PGL2

⊔MOp0,w0

PGL2.

By Theorem 9.3.6, we have

MOp0,wPGL2

' Conn(Ω−ρ)w(ρ)−ρD

(here we switch from Ωρ to Ω−ρ, as before).The stratum MOp0,1

PGL2for w = 1 fibers over the subspace of regular opers

OpPGL2(D) ⊂ Opnilp

PGL2(those with zero residue, v−1 = 0). The corresponding

space of connections Conn(Ω−ρ)w(ρ)−ρD is the space of regular connections on

the line bundle Ω−1/2 on D. Using our coordinate t, we represent such aconnection as an operator

∇ = ∂t + u(t), u(t) =∑n≥0

untn.

The map MOp0,1PGL2

→ OpPGL2(D) is the Miura transformation

∂t + u(t) 7→ (∂t − u(t)) (∂t + u(t)) = ∂2t − v(t),

where

v(t) = u(t)2 − ∂tu(t). (10.4.19)

These formulas were previously obtained in Section 8.2.2, except that we havenow rescaled u(t) by a factor of 2 to simplify our calculations.

The fiber of the Miura transformation Conn(Ω−1/2)D → OpPGL2(D) at

v(t) ∈ C[[t]] ' OpPGL2(D) consists of all solutions u(t) ∈ C[[t]] of the Riccati


equation (10.4.19). This equation amounts to an infinite system of algebraicequations on the coefficients un of u(t), which may be solved recursively.These equations read

nun = −vn−1 +∑

i+j=n−1

uiuj , n ≥ 1.

Thus, we see that all coefficients un, n ≥ 1, are uniquely determined by theseequations for each choice of u0, which could be an arbitrary complex number.Therefore the fiber of this map is isomorphic to the affine line, as expected.

For each solution u(t) ∈ C[[t]] of these equations we have the Wakimotomodule Wu(t) as defined in Corollary 6.1.5. According to Theorem 8.3.3, thecenter Z(sl2) acts on Wu(t) via the central character corresponding to thePGL2-oper χ corresponding to the projective connection ∂2

t − v(t). Thus, weobtain a family of Wakimoto modules in the category sl2 -modχ parameterizedby the affine line. These are the objects corresponding to the skyscrapersheaves supported at the points of MOp0,1

PGL2.

Now we consider the points of the other stratum

MOp0,w0PGL2

' Conn(Ω−1/2)−1D .

The corresponding connections on Ω−1/2 have regular singularity at 0 ∈ D

with the residue −1 (corresponding to −2ρ). With respect to our coordinatet they correspond to the operators

∇ = ∂t + u(t), u(t) =∑

n≥−1

untn, u−1 = −1.

The Miura transformation Conn(Ω−1/2)−1D → Opnilp

PGL2is again given by for-

mula (10.4.19). Let us determine the fiber of this map over a nilpotent opercorresponding to the projective connection ∂2

t −v(t). We obtain the followingsystem of equations on the coefficients un of u(t):

(n+ 2)un = −vn−1 +∑

i+j=n−1;i,j≥0

uiuj , n ≥ 0 (10.4.20)

(here we use the fact that u−1 = −1). This system may be solved re-cursively, but now u0 is determined by the first equation, and so there isno free parameter as in the previous case. Thus, we find that the mapConn(Ω−1/2)−1

D → OpnilpPGL2

is an isomorphism, as expected.If v−1 = 0, and so our oper χ = ∂2 − v(t) is regular, then we obtain an

additional point in the fiber of the Miura transformation over χ. Therefore wefind that the set of points in the fiber over a regular oper χ is indeed in bijectionwith the set of points of the projective line. The Wakimoto module Wu(t),

where u(t) is as above, is the object of the category sl2 -modI0

χ correspondingto the extra point represented by u(t). Thus, for each regular oper χ we


obtain a family of Wakimoto modules in sl2 -modI0

χ parameterized by P1, asexpected.

Now suppose that v−1 6= 0, so the residue of our oper χ is non-zero, andhence it has non-trivial unipotent monodromy. Then the set-theoretic fiber ofthe Miura transformation over χ consists of one point. We have one Wakimotomodule Wu(t) with u(t) = −1/t + . . . corresponding to this point. However,the fiber of MOp0

PGL2is a double point. Can we see this doubling effect from

the point of view of the Miura transformation?It turns out that we can. Let us consider the equation (10.4.19) with v−1 6= 0

and u(t) =∑

n∈Z untn, where un ∈ C[ε]/(ε2). It is easy to see that the

general solution to this equation such that u(t) mod ε is the above solutionured(t) = −1/t+

∑n≥0 ured,nt

n, has the form

ua(t) = ured(t) +aε

t2+ aε

∑n≥−1

δntn,

where a is an arbitrary complex number and δn ∈ C are uniquely determinedby (10.4.19). Thus, we obtain that the scheme-theoretic fiber of the Miuratransformation Conn(Ω−ρ)D× → Opnilp

PGL2at χ of the above form (with v−1 6=

0) is a double point, just like the fiber of MOp0PGL2

.This suggests that in general the scheme-theoretic fiber of the Miura trans-

formation over a nilpotent oper χ ∈ OpnilpLG

is also isomorphic to the non-reduced Springer fiber SpResF (χ). In [FG2] it was shown that this is indeedthe case.

The existence of the above one-parameter family of solutions ua(t), a ∈ C,of (10.4.19) over C[ε]/(ε2) means that there is a one-dimensional space ofself-extensions of the Wakimoto module Wured(t). Recall that this moduleis realized in the vector space Msl2 , the Fock module of the Weyl algebragenerated by an, a

∗n, n ∈ Z. This extension is realized in Msl2 ⊗ C[ε]/(ε2).

The action of the Lie algebra sl2 is given by the same formulas as before(see Section 6.1.2), except that now bn, n ∈ Z, acts as the linear operator onC[ε]/(ε2) corresponding to the nth coefficient ua,n of ua(t).

In fact, it follows from Theorem 10.4.5 (or from an explicit calculation) thatthe category sl2 -modI0

χ corresponding to χ ∈ OpnilpPGL2

with non-zero residuehas a unique irreducible object; namely, Wured(t). We note that this module isisomorphic to the quotient of the Verma module M−2 by the central charactercorresponding to χ. This module has a non-trivial self-extension described bythe above formula, and the category sl2 -modI0

χ is equivalent to the categoryQCoh(Spech0), where h0 = C[ε]/(ε2).

Now let us consider the category sl2 -modI0

χ , where χ is a PGL2-oper cor-responding to the operator ∂2

t − v(t), which is regular at t = 0 (and so itsresidue is v−1 = 0). We have constructed a family of objects Wu(t) of thiscategory, where u(t) are the solutions of the Riccati equation (10.4.19), which


are in one-to-one correspondence with points of P1 (for all but one of themu(t) is regular at t = 0 and the last one has the form u(t) = −1/t+ . . .). Whatis the structure of these Wakimoto modules? It is not difficult to see directlythat each of them is an extension of the same two irreducible sl2-modules,which are in fact the only irreducible objects of the category sl2 -modI0

χ . Thefirst of them is the module V0(χ) introduced in Section 10.3.3, and the secondmodule is obtained from V0(χ) by the twist with the involution of sl2 definedon the Kac–Moody generators (see Appendix A.5) by the formula

e0 ↔ e1, h0 ↔ h1, f0 ↔ f1.

We denote this module by V−2(χ). While V0(χ) is generated by a vectorannihilated by the “maximal compact” Lie subalgebra k0 = sl2[[t]], V−2(χ) isgenerated by a vector annihilated by the Lie subalgebra k′0, which is obtainedby applying the above involution to sl2[[t]]. These two Lie subalgebras, k0 andk′0, are the two “maximal compact” Lie subalgebras of sl2 up two conjugationby SL2((t)). In particular, the generating vector of V−2(χ) is annihilated bythe Lie algebra n+ and the Cartan subalgebra h acts on it according to theweight −2 (corresponding to −α). Thus, V−2(χ) is a quotient of the Vermamodule M−2.

We have a two-dimensional space of extensions Ext1(V−2(χ),V0(χ)). Sincethe extensions corresponding to vectors in Ext1(V−2(χ),V0(χ)) that are pro-portional are isomorphic as sl2-modules, we obtain that the isomorphismclasses of extensions are parameterized by points of P1. They are representedprecisely by our Wakimoto modules Wu(t). Thus, our P1, the fiber of theMiura transformation at a regular oper, acquires a useful interpretation asthe projectivization of the space of extensions of the two irreducible objectsof the category sl2 -modI0

χ .According to our conjectures, the derived categoryDb(sl2 -modχ)I0

is equiv-alent to the derived category of quasicoherent sheaves on the DG Springer fiberat 0, which is ΠTP1. The Wakimoto modules considered above correspondto the skyscraper sheaves. What objects of the category of quasi-coherentsheaves correspond to the irreducible sl2-modules V0(χ) and V−2(χ)? Theanswer is that V0(χ) corresponds to the trivial line bundle O on P1 andV−2(χ) corresponds to the complex O(−1)[1], i.e., the line bundle O(−1)placed in cohomological degree −1. The exact sequences in the category ofquasicoherent sheaves on P1

0→ O(−1)→ O → Op → 0,

where p ∈ P1 and Op is the skyscraper sheaf supported at p, become the exactsequences in the category sl2 -modI0

χ

0→ V0(χ)→Wu(t) → V−2(χ)→ 0,

where u(t) is the solution of (10.4.19) corresponding to the point p.

10.5 From local to global Langlands correspondence 353

Now we observe that in the category sl2 -modI0

χ there is a complete symme-try between the objects V0(χ) and V−2(χ), because they are related by an in-volution of sl2 which gives rise to an auto-equivalence of sl2 -modI0

χ . In partic-ular, we have Ext1(V0(χ),V−2(χ)) ' C2. But in the category Db(QCoh(P1))we have

Ext1(O,O(−1)[1]) = H2(P1,O(−1)) = 0.

This is the reason why the category Db(QCoh(P1)) cannot be equivalent tosl2 -modI0

χ . It needs to be replaced with the category Db(QCoh(ΠTP1)) ofquasicoherent sheaves on the DG Springer fiber ΠTP1. This is a good illus-tration of the necessity of using the DG Springer fibers in the equivalence(10.4.14).

10.5 From local to global Langlands correspondence

We now discuss the implications of the local Langlands correspondence studiedin this chapter for the global geometric Langlands correspondence.

The setting of the global correspondence that was already outlined in Sec-tion 1.1.6. We recall that in the classical picture we start with a smoothprojective curve X over Fq. Denote by F the field Fq(X) of rational functionson X. For any closed point x of X we denote by Fx the completion of F at xand by Ox its ring of integers. Thus, we now have a local field Fx attached toeach point ofX. The ring A = AF of adeles of F is by definition the restrictedproduct of the fields Fx, where x runs over the set |X| of all closed points ofX (the meaning of the word “restricted” is explained in Section 1.1.6). It hasa subring O =

∏x∈X Ox.

Let G be a (split) reductive group over Fq. Then for each x ∈ X we havea group G(Fx). Their restricted product over all x ∈ X is the group G(AF ).Since F ⊂ AF , the group G(F ) is a subgroup of G(AF ).

On one side of the global Langlands correspondence we have homomor-phisms σ : WF → LG satisfying some properties (or perhaps, some morerefined data, as in [Art]). We expect to be able to attach to each σ an auto-morphic representation π of GLn(AF ).† The word “automorphic” means,roughly, that the representation may be realized in a reasonable space of func-tions on the quotient GLn(F )\GLn(A) (on which the group GLn(A) acts fromthe right). We will not try to make this precise. In general, we expect not onebut several automorphic representations assigned to σ, which are the globalanalogues of the L-packets discussed above (see [Art]). Another complicationis that the multiplicity of a given irreducible automorphic representation inthe space of functions on GLn(F )\GLn(A) may be greater than one. We will

† In this section, by abuse of notation, we will use the same symbol to denote a represen-tation of a group and the vector space underlying this representation.


mostly ignore all of these issues here, as our main interest is in the geometrictheory (note that these issues do not arise if G = GLn).

An irreducible automorphic representation may always be decomposed asthe restricted tensor product

⊗′x∈X πx, where each πx is an irreducible repre-

sentation of G(Fx). Moreover, for all by finitely many x ∈ X the factor πx isan unramified representation of G(Fx): it contains a non-zero vector invari-ant under the maximal compact subgroup K0,x = G(Ox) (see Section 10.3.1).Let us choose such a vector vx ∈ πx (it is unique up to a scalar). The word“restricted” means that we consider the span of vectors of the form ⊗x∈Xux,where ux ∈ πx and ux = vx for all but finitely many x ∈ X.

An important property of the global Langlands correspondence is its com-patibility with the local one. We can embed the Weil group WFx

of each ofthe local fields Fx into the global Weil group WF . Such an embedding is notunique, but it is well-defined up to conjugation in WF . Therefore an equiv-alence class of σ : WF → LG gives rise to a well-defined equivalence class ofσx : WFx

→ LG. We will impose the condition on σ that for all but finitelymany x ∈ X the homomorphism σx is unramified (see Section 10.3.1).

By the local Langlands correspondence, to σx one can attach an equiva-lence class of irreducible smooth representations πx of G(Fx).† Moreover, anunramified σx will correspond to an unramified irreducible representation πx.The compatibility between local and global correspondences is the statementthat the automorphic representation of G(A) corresponding to σ should beisomorphic to the restricted tensor product

⊗′x∈X πx. Schematically, this is

represented as follows:

σglobal←→ π =

⊗x∈X

′πx

σxlocal←→ πx.

In this section we discuss the analogue of this local-to-global principle inthe geometric setting and the implications of our local results and conjecturesfor the global geometric Langlands correspondence. We focus in particular onthe unramified and tamely ramified Langlands parameters.

10.5.1 The unramified case

An important special case is when σ : WF → LG is everywhere unramified.Then for each x ∈ X the corresponding homomorphism σx : WFx

→ LG isunramified, and hence corresponds, as explained in Section 10.3.1, to a semi-simple conjugacy class γx in LG, which is the image of the Frobenius elementunder σx. This conjugacy class in turn gives rise to an unramified irreducible

† Here we are considering `-adic homomorphisms from the Weil group WFx to LG, andtherefore we do not need to pass from the Weil group to the Weil–Deligne group.


representation πx of G(Fx) with a unique, up to a scalar, vector vx such thatG(Ox)vx = vx. The spherical Hecke algebra H(G(Fx), G(Ox)) ' Rep LG actson this vector according to formula (10.3.5):

HV,x ? vx = Tr(γx, V )vx, [V ] ∈ Rep LG. (10.5.1)

The tensor product v =⊗

x∈X vx of these vectors is a G(O)-invariant vectorin π =

⊗′x∈X πx, which, according to the global Langlands conjecture, is

automorphic. This means that π is realized in the space of functions onG(F )\G(AF ). In this realization the vector v corresponds to a right G(O)-invariant function on G(F )\G(AF ), or, equivalently, a function on the doublequotient

G(F )\G(AF )/G(O). (10.5.2)

Thus, an unramified global Langlands parameter σ gives rise to a functionon (10.5.2). This function is the automorphic function corresponding toσ. We denote it by fπ. Since it corresponds to a vector in an irreduciblerepresentation π of G(AF ), the entire representation π may be reconstructedfrom this function. Thus, we do not lose any information by passing from π

to fπ.Since v ∈ π is an eigenvector of the Hecke operators, according to formula

(10.5.1), we obtain that the function fπ is a Hecke eigenfunction on thedouble quotient (10.5.2). In fact, the local Hecke algebras H(G(Fx), G(Ox))act naturally (from the right) on the space of functions on (10.5.2), and fπ isan eigenfunction of this action. It satisfies the same property (10.5.1).

To summarize, the unramified global Langlands correspondence in the clas-sical setting may be viewed as a correspondence between unramified homo-morphisms σ : WF → LG and Hecke eigenfunctions on (10.5.2) (some irre-ducibility condition on σ needs to be added to make this more precise, butwe will ignore this).

What should be the geometric analogue of this correspondence when X isa complex algebraic curve?

As explained in Section 1.2.1, the geometric analogue of an unramifiedhomomorphism WF → LG is a homomorphism π1(X)→ LG, or, equivalently,since X is assumed to be compact, a holomorphic LG-bundle on X witha holomorphic connection (it automatically gives rise to a flat connection,see Section 1.2.3). The global geometric Langlands correspondence shouldtherefore associate to a flat holomorphic LG-bundle on X a geometric objecton a geometric version of the double quotient (10.5.2). As we argued inSection 1.3.2, this should be a D-module on an algebraic variety whose set ofpoints is (10.5.2).

Now, it is known that (10.5.2) is in bijection with the set of isomorphismclasses of G-bundles on X. This key result is due to A. Weil (see, e.g., [F7]).This suggests that (10.5.2) is the set of points of the moduli space of G-


bundles on X. Unfortunately, in general this is not an algebraic variety, butan algebraic stack, which locally looks like a quotient of an algebraic varietyby an action of an algebraic group. We denote it by BunG. The theory ofD-modules has been developed in the setting of algebraic stacks like BunG in[BD1], and so we can use it for our purposes. Thus, we would like to attachto a flat holomorphic LG-bundle E on X a D-module AutE on BunG. ThisD-module should satisfy an analogue of the Hecke eigenfunction condition,which makes it into a Hecke eigensheaf with eigenvalue E. This notion isspelled out in [F7] (following [BD1]), where we refer the reader for details.

Thus, the unramified global geometric Langlands correspondence may besummarized by the following diagram:

flat LG-bundles on X −→ Hecke eigensheaves on BunG

E 7→ AutE

(some irreducibility condition on E should be added here).The unramified global geometric Langlands correspondence has been proved

in [FGV2, Ga2] for G = GLn and an arbitrary irreducible GLn-local systemE, and in [BD1] for an arbitrary simple Lie group G and those LG-localsystems which admit the structure of an oper. In this section we discuss thesecond approach.

To motivate it, we ask the following question:

How to relate this global correspondence to the local geometricLanglands correspondence discussed above?

The key element in answering this question is a localization functor fromg-modules to (twisted) D-modules on BunG. In order to construct this functorwe note that for a simple Lie group G the moduli stack BunG has anotherrealization as a double quotient. Namely, let x be a point of X. Denote byOx the completed local ring and by Kx its field of fractions. Let G(Kx) theformal loop group corresponding to the punctured disc D×x = SpecKx aroundx and by Gout the group of algebraic maps X\x → G, which is naturally asubgroup of G(Kx). Then BunG is isomorphic to the double quotient

BunG ' Gout\G(Kx)/G(Ox). (10.5.3)

This is a “one-point” version of (10.5.2). This is not difficult to prove at thelevel of C-points, but the isomorphism is also true at the level of algebraicstacks (see [BL, DrSi]).

The localization functor that we need is a special case of the followinggeneral construction. Let (g,K) be a Harish-Chandra pair and g -modK thecategory of K-equivariant g-modules (see Section 10.2.2). For a subgroup


H ⊂ G let DH\G/K -mod be the category of D-modules on H\G/K. Thenthere is a localization functor [BB, BD1] (see also [F7, FB])

∆ : g -modK → DH\G/K -mod .

Now let g be a one-dimensional central extension of g, which becomes trivialwhen restricted to the Lie subalgebras LieK and LieH. Suppose that thiscentral extension can be exponentiated to a central extension G of the corre-sponding Lie group G. Then we obtain a C×-bundle H\G/K over H\G/K.Let L be the corresponding line bundle. Let DL be the sheaf of differentialoperators acting on L. Then we have a functor

∆L : g -modK → DL -mod .

In our case we take the Lie algebra gκc,x, the critical central extension ofthe loop algebra g ⊗ Kx and the subgroups K = G(Ox) and H = Gout ofG(Kx). It is known (see, e.g., [BD1]) that the corresponding line bundle L isthe square root K1/2 of the canonical line bundle on BunG.† We denote thecorresponding sheaf of twisted differential operators on BunG by Dκc

. Thenwe have the localization functor

∆κc,x : gκc,x -modG(Ox) → Dκc -mod .

We restrict it further to the category gκc,x -modG(Ox)χx

for some

χx ∈ OpLG(D×x ).

According to Lemma 10.3.2, this category is non-zero only if χx ∈ OpλLG(Dx)

for some λ ∈ P+. Suppose for simplicity that λ = 0 and so χx ∈ OpLG(Dx).In this case, by Theorem 10.3.3, the category gκc,x -modG(Ox)

χxis equivalent to

the category of vector spaces. Its unique, up to an isomorphism, irreducibleobject is the quotient V0(χx) of the vacuum module

V0,x = Indbgκc,x

g(Ox)⊕C1 C

by the central character corresponding to χx. Hence it is sufficient to deter-mine ∆κc,x(V0(χx)).

The following theorem is due to Beilinson and Drinfeld [BD1].

Theorem 10.5.1 (1) The Dκc-module ∆κc,x(V0(χx)) is non-zero if and only

if there exists a global Lg-oper on X, χ ∈ OpLG(X) such that χx ∈ OpLG(Dx)is the restriction of χ to Dx.

(2) If this holds, then ∆κc,x(V0(χx)) depends only on χ and is independentof the choice of x in the sense that for any other point y ∈ X, if χy = χ|Dy

,then ∆κc,x(V0(χx)) ' ∆κc,y(V0(χy)).

† Recall that by our assumption G is simply-connected. In this case there is a uniquesquare root.


(3) For any χ = (F ,∇,FLB) ∈ OpLG(X) the Dκc-module ∆κc,x(V0(χx)) isa non-zero Hecke eigensheaf with the eigenvalue Eχ = (F,∇).

Thus, for any χ ∈ OpLG(X), the Dκc -module ∆κc,x(V0(χx)) is the sought-after Hecke eigensheaf AutEχ

corresponding to the LG-local system Eχ underthe global geometric Langlands correspondence (10.5.1).† For an outline ofthe proof of this theorem from [BD1], see [F7]. The key point is that theirreducible modules V0(χy), where χy = χ|Dy

, corresponding to all pointsy ∈ X, are Hecke eigenmodules as we saw in Section 10.3.5.

A drawback of this construction is that not all LG-local systems on X admitthe structure of an oper. In fact, under our assumption that LG is a group ofthe adjoint type, the LG-local systems, or flat bundles (F ,∇), on a smoothprojective curve X that admit an oper structure correspond to a unique LG-bundle Foper on X (described in Section 4.2.4). Moreover, for each such flatbundle (Foper,∇) there is a unique LB-reduction Foper,LB satisfying the opercondition. Therefore OpG(D) is the fiber of the forgetful map

LocLG(X)→ BunLG, (F ,∇) 7→ F ,

over the oper bundle Foper.For a LG-local system E = (F ,∇) for which F 6= Foper, the above con-

struction may be modified as follows (see the discussion in [F7] based on anunpublished work of Beilinson and Drinfeld). Suppose that we can choosean LB-reduction FLB satisfying the oper condition away from a finite set ofpoints y1, . . . , yn and such that the restriction χyi

of the corresponding operχ on X\y1, . . . , yn to D×yi

belongs to OpλiLG

(Dyi) ⊂ OpLG(D×yi) for some

λi ∈ P+. Then we can construct a Hecke eigensheaf corresponding to E byapplying a multi-point version of the localization functor to the tensor productof the quotients Vλi(χyi) of the Weyl modules Vλi,yi (see [F7]).

The main lesson of Theorem 10.5.1 is that in the geometric setting thelocalization functor gives us a powerful tool for converting local Langlandscategories, such as gκc,x -modG(Ox)

χx, into global categories of Hecke eigen-

sheaves. Therefore the emphasis shifts to the study of local categories ofgκc,x-modules. We can learn a lot about the global categories by studyingthe local ones. This is a new phenomenon which does not have any obviousanalogues in the classical Langlands correspondence.

In the case at hand, the category gκc,x -modG(Ox)χx

turns out to be verysimple: it has a unique irreducible object, V0(χx). That is why it is sufficientto consider its image under the localization functor, which turns out to bethe desired Hecke eigensheaf AutEχ . For general opers, with ramification,the corresponding local categories are more complicated, as we have seen

† More precisely, AutEχ is the D-module ∆κc,x(V0(χx))⊗K−1/2, but here and below we

will ignore the twist by K1/2.


above, and so are the corresponding categories of Hecke eigensheaves. Wewill consider examples of these categories in the next section.

10.5.2 Global Langlands correspondence with tame ramification

Let us first consider ramified global Langlands correspondence in the clas-sical setting. Suppose that we are given a homomorphism σ : WF → LG

that is ramified at finitely many points y1, . . . , yn of X. Then we expectthat to such σ corresponds an automorphic representation

⊗′x∈X πx (more

precisely, an L-packet of representations). Here πx is still unramified for allx ∈ X\y1, . . . , yn, but is ramified at y1, . . . , yn, i.e., the space of G(Oyi

)-invariant vectors in πyi

is zero. In particular, consider the special case wheneach σyi : WFyi

→ LG is tamely ramified (see Section 10.4.1 for the definition).Then, according to the results presented in Section 10.4.1, the correspondingL-packet of representations of G(Fyi

) contains an irreducible representationπyi with non-zero invariant vectors with respect to the Iwahori subgroup Iyi .Let us choose such a representation for each point yi.

Consider the subspacen⊗

i=1

πIyiyi ⊗

⊗x6=yi

vx ⊂⊗x∈X

′πx, (10.5.4)

where vx is a G(Ox)-vector in πx, x 6= yi, i = 1, . . . , n. Then, because⊗′x∈X πx is realized in the space of functions on G(F )\G(AF ), we obtain

that the subspace (10.5.4) is realized in the space of functions on the doublequotient

G(F )\G(AF )/n∏

i=1

Iyi ×∏

x6=yi

G(Ox). (10.5.5)

The spherical Hecke algebras H(G(Fx), G(Ox)), x 6= yi, act on the subspace(10.5.4), and all elements of (10.5.4) are eigenfunctions of these algebras (theysatisfy formula (10.5.1)). At the points yi, instead of the action of the com-mutative spherical Hecke algebra H(G(Fyi

), G(Oyi), we have the action of the

non-commutative affine Hecke algebra H(G(Fyi), Iyi

). Thus, we obtain a sub-space of the space of functions on (10.5.5), which consists of Hecke eigenfunc-tions with respect to the spherical Hecke algebras H(G(Fx), G(Ox)), x 6= yi,and which realize a module over

⊗ni=1H(G(Fyi), Iyi) (which is irreducible,

since each πyi is irreducible).This subspace encapsulates the automorphic representation

⊗′x∈X πx the

way the automorphic function fπ encapsulates an unramified automorphicrepresentation. The difference is that in the unramified case the functionfπ spans the one-dimensional space of invariants of the maximal compactsubgroup G(O) in

⊗′x∈X πx, whereas in the tamely ramified case the subspace

(10.5.4) is in general a multi-dimensional vector space.


Now let us see how this plays out in the geometric setting. As we discussedbefore, the analogue of a homomorphism σ : WF → LG tamely ramified atpoint y1, . . . , yn ∈ X is now a local system E = (F ,∇), where F a LG-bundleF on X with a connection ∇ that has regular singularities at y1, . . . , yn andunipotent monodromies around these points. We will call such a local systemtamely ramified at y1, . . . , yn. What should the global geometric Langlandscorrespondence attach to such a local system? It is clear that we need tofind a geometric object replacing the finite-dimensional vector space (10.5.4)realized in the space of functions on (10.5.5).

Just as (10.5.2) is the set of points of the moduli stack BunG of G-bundles,the double quotient (10.5.5) is the set of points of the moduli stack BunG,(yi)

of G-bundles on X with parabolic structures at yi, i = 1, . . . , n. By defini-tion, a parabolic structure of a G-bundle P at y ∈ X is a reduction of the fiberPy of P at y to a Borel subgroup B ⊂ G. Therefore, as before, we obtain thata proper replacement for (10.5.4) is a category of D-modules on BunG,(yi). Asin the unramified case, we have the notion of a Hecke eigensheaf on BunG,(yi).But because the Hecke functors are now defined using the Hecke correspon-dences over X\y1, . . . , yn (and not over X as before), an “eigenvalue” of theHecke operators is now an LG-local system on X\y1, . . . , yn (rather thanon X). Thus, we obtain that the global geometric Langlands correspondencenow should assign to a LG-local system E on X tamely ramified at the pointsy1, . . . , yn a category AutE of D-modules on BunG,(yi) with the eigenvalueE|X\y1,...,yn,

E 7→ AutE .

We will construct these categories using a generalization of the localizationfunctor we used in the unramified case (see [FG2]). For the sake of notationalsimplicity, let us assume that our LG-local system E = (F ,∇) is tamelyramified at a single point y ∈ X. Suppose that this local system on X\yadmits the structure of a LG-oper χ = (F ,∇,FLB) whose restriction χy tothe punctured disc D×y belongs to the subspace Opnilp

LG(Dy) of nilpotent LG-

opers.For a simple Lie group G, the moduli stack BunG,y has a realization anal-

ogous to (10.5.3):

BunG,y ' Gout\G(Ky)/Iy.

Let Dκc,Iy be the sheaf of twisted differential operators on BunG,y acting onthe line bundle corresponding to the critical level (it is the pull-back of thesquare root of the canonical line bundle K1/2 on BunG under the naturalprojection BunG,y → BunG). Applying the formalism of the previous section,we obtain a localization functor

∆κc,Iy : gκc,y -modIy → Dκc,Iy -mod .


However, in order to make contact with the results obtained above we alsoconsider the larger category gκc,y -modI0

y of I0y -equivariant modules, where

I0y = [Iy, Iy].Set

Bun′G,y = Gout\G(Ky)/I0y ,

and let Dκc,I0y

be the sheaf of twisted differential operators on Bun′G,y actingon the pull-back of the line bundle K1/2 on BunG. Let Dκc,I0

y-mod be the

category of Dκc,I0y-modules. Applying the general formalism, we obtain a

localization functor

∆κc,I0y

: gκc,y -modI0y → Dκc,I0

y-mod . (10.5.6)

We note that a version of the categorical affine Hecke algebra H(G(Ky), Iy)discussed in Section 10.4.5 naturally acts on the derived categories of theabove categories, and the functors ∆κc,Iy and ∆κc,I0

yintertwine these actions.

Equivalently, one can say that this functor intertwines the corresponding ac-tions of the affine braid group associated to LG on the two categories (as in[B]).

We now restrict the functors ∆κc,Iy and ∆κc,I0y

to the subcategories

gκc,y -modIyχy

and gκc,y -modI0

yχy , respectively. By using the same argument

as in [BD1], we obtain the following analogue of Theorem 10.5.1.

Theorem 10.5.2 Fix χy ∈ OpnilpLG

(Dy) and let M be an object of the category

gκc,y -modIyχy

(resp. gκc,y -modI0

yχy). Then:

(1) ∆κc,Iy (M) = 0 (resp., ∆κc,I0y(M) = 0) unless χy is the restriction of a

regular oper χ = (F ,∇,FLB) on X\y to D×y .(2) In that case ∆κc,y(M) (resp., ∆κc,I0

y(M)) is a Hecke eigensheaf with

the eigenvalue Eχ = (F ,∇).

Thus, we obtain that if χy = χ|D×y, then the image of any object of

gκc,y -modIyχy

under the functor ∆κc,Iy belongs to the category AutIy

Eχof Hecke

eigensheaves on BunG,y. Now consider the restriction of the functor ∆κc,I0y

to gκc,y -modI0

yχy . As discussed in Section 10.4.3, the category gκc,y -mod

I0y

χy

coincides with the corresponding category gκc,y -modIy,mχy

of Iy-monodromic

modules. Therefore the image of any object of gκc,y -modI0

yχy under the func-

tor ∆κc,I0y

belongs to the subcategory Dmκc,I0

y-mod of Dκc,I0

y-mod whose ob-

jects admit an increasing filtration such that the consecutive quotients arepull-backs of Dκc,Iy

-modules from BunG,y. Such Dκc,I0y–modules are called

monodromic.Let AutIy,m

Eχbe the subcategory of Dm

κc,I0y-mod whose objects are Hecke

eigensheaves with eigenvalue Eχ.


Thus, we obtain the functors

∆κc,Iy:gκc,y -modIy

χy→ AutIy

Eχ, (10.5.7)

∆κc,I0y

:gκc,y -modI0

yχy → Aut

Iy,mEχ

. (10.5.8)

It is tempting to conjecture (see [FG2]) that these functors are equivalencesof categories, at least for generic χ. Suppose that this is true. Then we mayidentify the global categories AutIy

Eχand AutIy,m

Eχof Hecke eigensheaves on

BunG,Iy and Bun′G,I0y

with the local categories gκc,y -modIyχy

and gκc,y -modI0

yχy ,

respectively. Therefore we can use our results and conjectures on the local

Langlands categories, such as gκc,y -modI0

yχy , to describe the global categories

of Hecke eigensheaves on the moduli stacks of G-bundles on X with parabolicstructures.

Recall that we have the following conjectural description of the derived cat-egory of I0

y -equivariant modules, Db(gκc,y -modχy )I0y (see formula (10.4.14)):

Db(gκc,y -modχy )I0y ' Db(QCoh(Sp

DG

Res(χy))). (10.5.9)

The corresponding Iy-equivariant version is

Db(gκc,y -modχy )Iy ' Db(QCoh(SpDGRes(χy))), (10.5.10)

where we replace the non-reduced DG Springer fiber by the reduced one:

OSpDGu

= O eN L⊗OLg

Cu.

If the functors (10.5.7), (10.5.8) are equivalences, then by combining themwith (10.5.9) and (10.5.10), we obtain the following conjectural equivalencesof categories:

Db(AutIy

Eχ) ' Db(QCoh(SpDG

Res(χy))), (10.5.11)

Db(AutIy,mEχ

) ' Db(QCoh(SpDG

Res(χy))). (10.5.12)

In other words, the derived category of a global Langlands category (mon-odromic or not) corresponding to a local system tamely ramified at y ∈ X

is equivalent to the derived category of quasicoherent sheaves on the DGSpringer fiber of its residue at y (non-reduced or reduced).

Again, these equivalences are supposed to intertwine the natural actions onthe above categories of the categorical affine Hecke algebra H(G(Ky), Iy) (or,equivalently, the affine braid group associated to LG).

The categories appearing in (10.5.11), (10.5.12) actually make sense foran arbitrary LG-local system E on X tamely ramified at y. It is therefore


tempting to conjecture that these equivalences still hold in general:

Db(AutIy

E ) ' Db(QCoh(SpDGRes(E))), (10.5.13)

Db(AutIy,mE ) ' Db(QCoh(Sp

DG

Res(E))). (10.5.14)

The corresponding localization functors may be constructed as follows. Sup-pose that we can represent a local system E on X with tame ramification at yby an oper χ on the complement of finitely many points y1, . . . , yn, whose re-striction to D×yi

belongs to OpλiLG

(Dyi) ⊂ OpLG(D×yi) for some λi ∈ P+. Then,

in the same way as in the unramified case, we construct localization functors

from gκc,y -modIyχy

to AutIy

E and from gκc,y -modI0

yχy to AutIy,m

E (here, as be-fore, χy = χ|D×

y), and this leads us to the conjectural equivalences (10.5.13),

(10.5.14).These equivalences may be viewed as the geometric Langlands correspon-

dence for tamely ramified local systems.The equivalences (10.5.13) also have family versions in which we allow E to

vary. It is analogous to the family version (10.4.13) of the local equivalences.As in the local case, in a family version we can avoid using DG schemes.

The above construction may be generalized to allow local systems tamelyramified at finitely many points y1, . . . , yn. The corresponding Hecke eigen-sheaves are then D-modules on the moduli stack of G-bundles on X withparabolic structures at y1, . . . , yn. Non-trivial examples of these Hecke eigen-sheaves arise already in genus zero. These sheaves were constructed explicitlyin [F3] (see also [F5, F6]), and they are closely related to the Gaudin integrablesystem.

10.5.3 Connections with regular singularities

So far we have only considered the categories of gκc-modules corresponding to

LG-opers χ on X, which are regular everywhere except at a point y ∈ X (orperhaps, at several points) and whose restriction χ|D×

yis a nilpotent oper in

OpnilpLG

(Dy). In other words, χ|D×y

is an oper with regular singularity at y withresidue $(−ρ). However, we can easily generalize the localization functor tothe categories of gκc -modules corresponding to LG-opers, which have regularsingularity at y with arbitrary residue.

Recall that in Section 9.4.3 for each oper χ ∈ OpRSLG(D)$(−λ−ρ) with regular

singularity and residue $(−λ−ρ) we have defined the category gκc -modI0

χ ofI0-equivariant gκc

-modules with central character χ. The case of λ = 0 is an“extremal” case when the category gκc -modI0

χ is most complicated. On theother “extreme” is the case of generic opers χ corresponding to a generic λ. Inthis case, as we saw in Section 9.4.3, the category gκc

-modI0

χ is quite simple:it contains irreducible objects Mw(λ+ρ)−ρ(χ) labeled by the Weyl group of g,


and each object of gκc -modI0

χ is a direct sum of these irreducible modules.Here Mw(λ+ρ)−ρ(χ) is the quotient of the Verma module Mw(λ+ρ)−ρ, w ∈W ,by the central character corresponding to χ.

For other values of λ the structure of gκc-modI0

χ is somewhere in-betweenthese two extreme cases.

Recall that we have a localization functor (10.5.6)

∆λκc,I0

y: gκc,y -modI0

y → Dκc,I0y-mod

from gκc,y -modI0

yχy to a category of D-modules on Bun′G,Iy

twisted by thepull-back of the line bundle K1/2 on BunG. We now restrict this functor

to the subcategory gκc,y -modI0

yχy where χy is a LG-oper on Dy with regular

singularity at y and residue $(−λ− ρ).Consider first the case when λ ∈ h∗ is generic. Suppose that χy extends

to a regular oper χ on X\y. One then shows in the same way as in Theo-

rem 10.5.2 that for any object M of gκc,y -modI0

yχy the corresponding Dκc,I0

y-

module ∆κc,I0y(M) is a Hecke eigensheaf with eigenvalue Eχ, which is the

LG-local system on X with regular singularity at y underlying χ (if χy can-not be extended to X\y, then ∆λ

κc,Iy(M) = 0, as before). Therefore we obtain

a functor

∆κc,I0y

: gκc,y -modI0

yχy → Aut

I0y

Eχ,

where AutI0y

Eχis the category of Hecke eigensheaves on Bun′G,Iy

with eigenvalueEχ.

Since we have assumed that the residue of the oper χy is generic, the mon-odromy of Eχ around y belongs to a regular semi-simple conjugacy class of LG

containing exp(2πiλ). In this case the category gκc,y -modI0

yχy is particularly

simple, as we have discussed above. We expect that the functor ∆κc,I0y

sets

up an equivalence between gκc,y -modI0

yχy and AutI

0y

Eχ.

We can formulate this more neatly as follows. For M ∈ LG let BM bethe variety of Borel subgroups containing M . Observe that if M is regularsemi-simple, then BM is a set of points which is in bijection with W . There-

fore our conjecture is that AutI0y

Eχis equivalent to the category QCoh(BM ) of

quasicoherent sheaves on BM , where M is a representative of the conjugacyclass of the monodromy of Eχ.

Consider now an arbitrary LG-local system E on X with regular singularityat y ∈ X whose monodromy around y is regular semi-simple. It is thentempting to conjecture that, at least if E is generic, this category has thesame structure as in the case when E has the structure of an oper, i.e., itis equivalent to the category QCoh(BM ), where M is a representative of theconjugacy class of the monodromy of E around y.


On the other hand, if the monodromy around y is unipotent, then BM isnothing but the Springer fiber Spu, where M = exp(2πiu). The corresponding

category AutI0y

E was discussed in Section 10.5.2 (we expect that it coincideswith AutIy,m

E ). Thus, we see that in both “extreme” cases – unipotent mon-odromy and regular semi-simple monodromy – our conjectures identify the de-

rived category of AutI0y

E with the derived category of the category QCoh(BM )

(where BM should be viewed as a DG scheme SpDG

u in the unipotent case).One is then led to conjecture, most ambitiously, that for any LG-local system

E on X with regular singularity at y ∈ X the derived category of AutI0y

E isequivalent to the derived category of quasicoherent sheaves on a suitable DGversion of the scheme BM , where M is a representative of the conjugacy classof the monodromy of E around y:

Db(AutI0y

E ) ' Db(QCoh(BDGM )).

This has an obvious generalization to the case of multiple ramification points,where on the right hand side we take the Cartesian product of the varietiesBDG

Micorresponding to the monodromies. Thus, we obtain a conjectural re-

alization of the categories of Hecke eigensheaves, whose eigenvalues are lo-cal systems with regular singularities, in terms of categories of quasicoherentsheaves.

Let us summarize: by using the representation theory of affine Kac–Moodyalgebras at the critical level we have constructed the local Langlands cate-gories corresponding to local Langlands parameters; namely, LG-local sys-tems on the punctured disc. We then applied the technique of localizationfunctors to produce from these local categories, the global categories of Heckeeigensheaves on the moduli stacks of G-bundles on a curve X with parabolicstructures. These global categories correspond to global Langlands parame-ters: LG-local systems on X with ramifications. We have used our results andconjectures on the structure of the local categories to investigate these globalcategories.

In this chapter we have explained how this works for unramified and tamelyramified local systems. But we expect that analogous methods may also beused to analyze local systems with arbitrary ramification. In this way therepresentation theory of affine Kac–Moody algebras may one day fulfill thedream of uncovering the mysteries of the geometric Langlands correspondence.

Appendix

A.1 Lie algebras

A Lie algebra is a vector space g with a bilinear form (the Lie bracket)

[·, ·] : g⊗ g −→ g,

which satisfies two additional conditions:

• [x, y] = − [y, x]• [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0.

The last of these conditions is called the Jacobi identity.Lie algebras are closely related to Lie groups. The tangent space at the

identity element of a Lie group naturally has the structure of a Lie algebra.So, thinking about Lie algebras is a way of linearizing problems coming fromthe theory of Lie groups.

A representation of a Lie algebra g, or equivalently, a g-module, is a pair(V, ρ) of a vector space V and a map ρ : g→ End(V ) such that

ρ([x, y]) = ρ(x)ρ(y)− ρ(y)ρ(x).

We will often abuse notation and call the representation V rather than spec-ifying the homomorphism ρ; however, ρ will always be implicit.

A.2 Universal enveloping algebras

A standard tool in the study of Lie algebras and their representations is theuniversal enveloping algebra. This is an associative algebra that can be con-structed from any Lie algebra and has the property that any module over theLie algebra can be regarded as a module over this associative algebra.

As we want to be able to multiply elements of g together in the associativealgebra it makes sense to look at the tensor algebra of g

T (g) = C⊕ g⊕ g⊗2 ⊕ g⊗3 ⊕ · · · .

If we have a representation (V, ρ) of g, we clearly want want the corresponding

366

A.2 Universal enveloping algebras 367

T (g)-module to be such that the action of T (g) restricts to (V, ρ) on the secondsummand. Since it should be a module over an associative algebra, we alsowant

ρ(g ⊗ h) = ρ(g)ρ(h).

However, the Lie algebra modules satisfy one additional relation

ρ([g, h]) = ρ(g)ρ(h)− ρ(h)ρ(g).

Therefore the elements [g, h] and g⊗h−h⊗ g of T (g) should act in the sameway. This suggests that we should identify [g, h] and g ⊗ h − h ⊗ g in thetensor algebra and this leads us to the following definition.

The universal enveloping algebra U(g) is the quotient of the tensoralgebra T (g) by the two sided ideal generated by elements of the form

g ⊗ h− h⊗ g − [g, h] .

Note that we have a natural map g→ U(g).It is called universal because it has the following universal property: If A is

any associative algebra (regarded naturally as a Lie algebra) and f : g→ A is aLie algebra homomorphism then it may be uniquely completed to the followingcommutative diagram:

U(g)xg −−−−→ A

QQ

QQs

There is a natural filtration (sometimes called the PBW filtration) on theuniversal enveloping algebra coming from the gradation on the tensor algebra.The ith term of the filtration is denoted by U(g)6i.

The structure of the universal enveloping algebra is very easy to work outin the case that g is abelian. In this case the generators for the ideal aresimply g ⊗ h− h⊗ g (as the bracket is zero). This is exactly the definition ofthe symmetric algebra generated by g. Hence we get

U(g) ∼= Sym(g).

In the case where g is not abelian, the situation is more complicated.Given any filtered algebra

0 = F6−1 ⊂ F60 ⊂ F61 ⊂ · · · ⊂ F

we define the associated graded algebra to be

grF =⊕i>0

F6i/F6i−1.

368 Appendix

This is naturally a graded algebra and considered as vector spaces we havegrF ∼= F , but non-canonically. Taking the associated graded algebra of aquotient is reasonably easy:

gr(F/I) = gr(F )/Symb(I)

where Symb is the symbol map. Let x be an element of a filtered algebraF . There is an integer i such that x ∈ F6i but x 6∈ F6i−1. The symbol of x isdefined to be the image of x in the quotient F6i/F6i−1. In other words, thesymbol map is picking out the leading term of x.

So, for the universal enveloping algebra we have

grU(g) = grT (g)/Symb(I).

As the tensor algebra is already graded, grT (g) ∼= T (g).

Lemma A.2.1 The ideal Symb(I) is generated by the symbols of the genera-tors for the ideal I.

The proof of this fact is the heart of most proofs of the Poincare–Birkhoff–Witt theorem. It fundamentally relies on the fact that the bracket satisfies theJacobi identity. If we were using an arbitrary bilinear form to define the idealI (e.g., if we used the ideal generated by a⊗ b− b⊗ a−B(a, b)), there wouldbe elements in Symb(I) not obtainable from the symbols of the generators.The symbols of the generators are easy to work out and are just g⊗h−h⊗g.The corresponding quotient is Sym(g). Hence we have shown that

grU(g) ∼= Sym(g)

for an arbitrary Lie algebra g.In other words, as vector spaces, U(g) and Sym(g) are isomorphic, but

non-canonically. This is often quoted in the following form.

Theorem A.2.2 (Poincare–Birkhoff–Witt) Let Ja, a = 1, . . . ,dim g, bean ordered basis for g (as a vector space), then the universal enveloping algebrahas a basis given by elements of the form Ja1 . . . Jan , where a1 ≤ . . . ≤ an.

Monomials of this form are often called lexicographically ordered. Oneof the main reasons for studying the universal enveloping algebras is that thecategory of U(g)-modules is equivalent to the category of g-modules (this isjust the statement that any representation of g can be given a unique structureof module over U(g)).

A.3 Simple Lie algebras

In this book we consider simple finite-dimensional complex Lie algebras. Here“simple” means that the Lie algebra has no non-trivial ideals. These Lie alge-

A.3 Simple Lie algebras 369

bras have been completely classified. They are in one-to-one correspondencewith the positive definite Cartan matrices A = (aij)i,j=1,...,`. These aresymmetrizable matrices with integer entries satisfying the following condi-tions: aii = 2, and for i 6= j we have aij ≤ 0 and aij = 0 if and only if aji 6= 0(see [K2] for more details). Such matrices, and hence simple Lie algebras, fitinto four infinite families and six exceptional ones. The families are called An,Bn, Cn, and Dn; the exceptional Lie algebras are called E6, E7, E8, F4 andG2.

A simple Lie algebra g admits a Cartan decomposition

g = n− ⊕ h⊕ n+.

Here h is a Cartan subalgebra, a maximal commutative Lie subalgebra ofg consisting of semi-simple elements (i.e., those whose adjoint action on g

is semi-simple). Its dimension is `, which is the rank of the Cartan matrixcorresponding to g; we will also call ` the rank of g. The Lie subalgebra n+

(resp., n−) is called the upper (resp., lower) nilpotent subalgebra. Under theadjoint action of h the nilpotent subalgebras decompose into eigenspaces asfollows

n± =⊕

α∈∆+

n±α,

where

n±α = x ∈ n± | [h, x] = ±α(h)x.

Here ∆+ ⊂ h∗ and ∆− = −∆+ are the sets of positive and negative roots±α of g, respectively, for which n±α 6= 0. In fact, all of these subspaces areone-dimensional, and we will choose generators eα and fα of nα and n−α,respectively.

It is known that ∆+ contains a subset of simple roots αi, i = 1, . . . , `,where ` = dim h is the rank of g. All other elements of ∆+ are linear combi-nations of the simple roots with non-negative integer coefficients. The simpleroots form a basis of h∗. Let hi, i = 1, . . . , `, be the basis of h that is uniquelydetermined by the formula

αj(hi) = aij .

These are the simple coroots.Let us choose non-zero generators ei = eαi (resp., fi = fαi) of the one-

dimensional spaces nαi(resp., n−αi

), i = 1, . . . , `. Then n+ (resp., n−) isgenerated by e1, . . . , e` (resp., f1, . . . , f`) subject to the following Serre re-lations:

(ad ei)−aij+1 · ej = 0, (ad fi)−aij+1 · fj = 0, i 6= j.

Here the aij ’s are the entries of the Cartan matrix of g.

370 Appendix

The Lie algebra g is generated by hi, hi, fi, i = 1, . . . , `, subject to the Serrerelations together with

[hi, ej ] = aijej , [hi, fj ] = −aijfj , [ei, fj ] = δi,jhi.

To satisfy the last set of relations we might need to rescale the ei’s (or thefi’s) by non-zero numbers.

For example, the Lie algebra An is the Lie algebra also called sln+1. Thisis the vector space of (n + 1) × (n + 1) matrices with zero trace and bracketgiven by [A,B] = AB−BA. This is the Lie algebra of the special linear groupSLn+1(C), which consists of (n + 1) × (n + 1) matrices with determinant 1.To see this, we write an elements of SLn+1(C) in the form In+1 + εM , whereε2 = 0 and see what restriction the determinant condition puts on M . Thisis exactly the trace 0 condition.

The Cartan subalgebra consists of diagonal matrices. The Lie algebras n+

and n− are the upper and lower nilpotent subalgebras, respectively, with thezeros on the diagonal. The generators ei, hi and fi are the matrices Ei,i+1,Eii − Ei+1,i+1, and Ei+1,i, respectively.

There are four lattices attached to g: the weight lattice P ⊂ h∗ consists oflinear functionals φ : h → C such that φ(hi) ∈ Z; the root lattice Q ⊂ P isspanned over Z by the simple roots αi, i = 1, . . . , `; the coweight lattice P ⊂ h

consists of elements on which the simple roots take integer values; and thecoroot lattice Q is spanned by the simple coroots hi, i = 1, . . . , `.

The lattice P is spanned by elements ωi ∈ h∗ defined by ωi(hj) = δi,j .The coroot lattice is spanned by elements ωi ∈ h defined by the formulaαi(ωi) = δi,j .

We note that P may be identified with the group of characters Hsc → C×of the Cartan subgroup Hsc of the connected simply-connected Lie group G

with the Lie algebra g (such as SLn), and P is identified with the group ofcocharacters C× → Hadj, where Hadj is the Cartan subgroup of the group ofadjoint type associated to G (with the trivial center, such as PGLn).

Let si, i = 1, . . . , `, be the linear operator (reflection) on h∗ defined by theformula

si(λ) = λ− 〈λ, αi〉αi.

These operators generate an action of the Weyl group W , associated to g,on h∗.

A.4 Central extensions

By a central extension of a Lie algebra g by an abelian Lie algebra a weunderstand a Lie algebra g that fits into an exact sequence of Lie algebras

0 −→ a −→ g −→ g −→ 0. (A.4.1)

A.5 Affine Kac–Moody algebras 371

Here a is a Lie subalgebra of g, which is central in g, i.e., [a, x] = 0 for alla ∈ a, x ∈ g.

Suppose that a ' C1 is one-dimensional with a generator 1. Choose asplitting of g as a vector space, g ' C1⊕ g. Then the Lie bracket on g givesrise to a Lie bracket on C1⊕ g such that C1 is central and

[x, y]new = [x, y]old + c(x, y)1,

where x, y ∈ g and c : g⊗ g→ C is a linear map. The above formula definesa Lie bracket on g if and only if c satisfies:

• c(x, y) = −c(y, x).• c(x, [y, z]) + c(y, [z, x]) + c(z, [x, y]) = 0.

Such expressions are called two-cocycles on g.Now suppose that we are given two extensions g1 and g2 of the form (A.4.1)

with a = C1 and a Lie algebra isomorphism φ between them which preservestheir subspaces C1. Choosing splittings ı1 : C1⊕g→ g1 and ı2 : C1⊕g→ g2

for these extensions, we obtain the two-cocycles c1 and c2. Define a linear mapf : g→ C as the composition of ı1, φ and the projection g2 → C1 induced byı2. Then it is easy to see that

c2(x, y) = c1(x, y) + f([x, y]).

Therefore the set (actually, a vector space) of isomorphism classes of one-dimensional central extensions of g is isomorphic to the quotient of the vectorspace of the two-cocycles on g by the subspace of those two-cocycles c forwhich

c(x, y) = f([x, y]), ∀x, y ∈ g,

for some f : g → C (such cocycles are called coboundaries). This quotientspace is the second Lie algebra cohomology group of g, denoted by H2(g,C).

A.5 Affine Kac–Moody algebras

If we have a complex Lie algebra g and a commutative and associative algebraA, then g⊗

CA is a Lie algebra if we use the bracket

[g ⊗ a, h⊗ b] = [g, h]⊗ ab.

What exactly is A? One can usually think of commutative, associative alge-bras as functions on some manifold M . Note that

g⊗C

Fun(M) = Map(M, g)

is just the Lie algebra of maps from M to g. This is naturally a Lie algebrawith respect to the pointwise Lie bracket (it coincides with the Lie bracketgiven by the above formula).

372 Appendix

Let us choose our manifold to be the circle, so we are thinking about mapsfrom the circle into the Lie algebra. We then have to decide what type ofmaps we want to deal with. Using smooth maps would require some analysis,whereas we would like to stay in the realm of algebra – so we use algebraic(polynomial) maps. We consider the unit circle embedded into the complexplane with the coordinate t, so that on the circle t = eiθ, θ being the angle.Polynomial functions on this circle are the same as Laurent polynomials in t.They form a vector space denoted by C[t, t−1]. Using this space of functions,we obtain the polynomial loop algebra

Lg = g⊗ C[t, t−1]

with the bracket given as above.It turns out that there is much more structure and theory about the cen-

trally extended loop algebras. As we saw above, these are classified by thesecond cohomology group H2(Lg,C), and for a simple Lie algebra g this isknown to be one-dimensional. The corresponding cocycles are given by for-mula (1.3.3). They depend on the invariant inner product on g denoted by κ.We will denote the central extension by gpol

κ .There is a formal version of the above construction where we replace the

algebra C[t, t−1] of Laurent polynomials by the algebra C((t)) of formal Laurentseries. This gives is the formal loop algebra g ⊗ C((t)) = g((t)). Its centralextensions are parametrized by its second cohomology group, which, with theappropriate continuity condition, is also a one-dimensional vector space if g

is a simple Lie algebra. The central extensions are again given by specifyingan invariant inner product κ on g. The Lie bracket is given by the sameformula as before – note that the residue is still well defined. We denote thecorresponding Lie algebra gκ. This is the affine Kac-Moody algebra associatedto g (and κ).

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Index

adele, 6admissible representation, 301affine flag variety, 346affine Grassmannian, 21, 296, 317, 318, 340affine Hecke algebra, see Hecke algebra,

affineaffine Kac–Moody algebra, 28, 371annihilation operator, 141associated graded algebra, 78, 367automorphic function, 355automorphic representation, 7, 353, 359

big cell, 129

Cartan decomposition, 108, 369Cartan matrix, 369Cartan subalgebra, 369Casimir element, 32center

at the critical level, 77, 117, 234of a vertex algebra, 76of the enveloping algebra, 120, 124

central character, 297central charge, 68central extension, 27, 370conformal dimension, 40conformal vector, 68congruence subgroup, 2, 18, 302connection, 11, 109

flat, 11, 109constructible sheaf, 23

`-adic, 22contragredient representation, 21convolution, 302coweight, 370creation operator, 141critical level, see level, critical

D-module, 24, 303, 317, 318, 347, 355DG scheme, 335disc, 68, 81, 90, 93

punctured, 17, 93Drinfeld–Sokolov reduction, 239dual Coxeter number, 64

equivariant module, 302exponent of a Lie algebra, 112

field, 40flag variety, 129flat G-bundle, see G-bundle, flatflat connection, see connection, flatflat vector bundle, see vector bundle, flatformal coordinate, 90formal delta-function, 39formal loop group, see loop groupFrobenius automorphism, 3, 318, 327, 339,

354fundamental group, 9

G-bundle, 14flat, 15

Galois group, 3, 9gauge transformation, 15, 109Grothendieck alteration, 194, 335

Harish-Chandra category, 303Harish-Chandra module, 303Harish-Chandra pair, 303Hecke algebra, 302

affine, 329affine, categorical, 346, 361categorical, 304spherical, 306, 340spherical, categorical, 317

Hecke eigenfunction, 355Hecke eigenmodule, 318, 320, 340Hecke eigensheaf, 356, 360Hecke eigenvector, 307, 355highest weight vector, 185holomorphic vector bundle, see vector

bundle, holomorphichorizontal section, 13

integrable representation, 19intertwining operator, 197invariant inner product, 28Iwahori subgroup, 302, 327

jet scheme, 84, 286

377

378 Index

Kac–Kazhdan conjecture, 190Killing form, 28, 63

L-packet, 6, 328, 353Langlands correspondence

global, 6, 8, 353global geometric, tamely ramified, 363global geometric, unramified, 356global, tamely ramified, 359global, unramified, 355local, 1, 4, 6, 354local geometric, 29local geometric, tamely ramified, 331,

337local geometric, unramified, 307, 310, 322local, tamely ramified, 327local, unramified, 306

Langlands dual group, 6, 100, 108, 117,296, 306, 319

Langlands dual Lie algebra, 117, 221level, 28

critical, 30, 37, 64, 99, 140, 161, 162, 166lexicographically ordered monomial, 368Lie algebra, 366

simple, 368local field, 1local system, 18, 297

`-adic, 22tamely ramified, 332, 342, 360trivial, 307, 341

locality, 41localization functor, 356, 360, 363, 365loop algebra

formal, 26, 372polynomial, 372

loop group, 9, 18, 20, 23, 26, 30, 295

maximal compact subgroup, 21, 302, 305Miura oper, 193, 226

generic, 226nilpotent, 258

Miura transformation, 229, 241, 349module over a Lie algebra, 366monodromic module, 334monodromy, 13, 17, 327, 332, 333, 339, 364

nilpotent cone, 86, 260, 330nilpotent Miura oper, see Miura oper,

nilpotentnilpotent oper, see oper, nilpotentnormal ordering, 37, 49

oper, 109, 297nilpotent, 248, 333regular, 310with regular singularity, 244

operator product expansion, 56

PBW filtration, see universal envelopingalgebra, filtration

Poincare–Birkhoff–Witt theorem, 368

principal G-bundle, see G-bundleprincipal gradation, 112projective connection, 96, 101, 107projective structure, 102punctured disc, see disc, punctured

reconstruction theorem, 51, 52reduction of a principal bundle, 104regular singularity, 16, 244, 360, 363relative position, 261

generic, 194representation of a Lie algebra, 366residue, 28

of a nilpotent oper, 253of an oper with regular singularity, 244

Riemann–Hilbert correspondence, 24root, 369

simple, 369

Satake isomorphism, 306categorical, 317

Schwarzian derivative, 98, 114screening operator, 192

of W-algebra, 219of the first kind, 200, 206of the second kind, 203, 208

Segal–Sugawara operator, 37, 62, 74, 87,98, 119, 171

smooth representation, 2, 19, 29, 178, 183space of states, 40spherical Hecke algebra, see Hecke algebra,

sphericalSpringer fiber, 261, 329, 348Springer variety, 260state-field correspondence, 41Steinberg variety, 330, 347symbol, 368

Tannakian formalism, 15, 317, 340torsor, 90transition function, 11translation operator, 40

universal enveloping algebra, 367completed, 34filtration, 367

unramified representation, 305

vacuum vector, 40vacuum Verma module, see Verma module,

vacuumvector bundle, 11

flat, 11holomorphic, 13

Verma module, 132, 185, 273contragredient, 133vacuum, 45

vertex algebra, 40commutative, 42conformal, 68quasi-conformal, 173

Index 379

vertex algebra homomorphism, 68vertex operator, 41vertex Poisson algebra, 221Virasoro algebra, 66

W-algebra, 220Wakimoto module, 169

generalized, 183of critical level, 166, 343, 351

weight, 370Weil group, 4, 7, 354Weil–Deligne group, 4, 306, 326, 339Weyl algebra, 138Weyl group, 370Weyl module, 279Wick formula, 147

Documents

[Edward Frenkel] Langlands Correspondence for Loop(BookZa.org)