University of Torontoblog.math.toronto.edu/GraduateBlog/files/2016/03/Thesis3.pdfAbstract The Equivariant Geometry of Nilpotent Orbits and Associated Varieties Peter Crooks Doctor

THE EQUIVARIANT GEOMETRY OF NILPOTENT ORBITS AND

ASSOCIATED VARIETIES

by

Peter Crooks

A thesis submitted in conformity with the requirementsfor the degree of Doctor of PhilosophyGraduate Department of Mathematics

University of Toronto

c© Copyright 2016 by Peter Crooks

AbstractThe Equivariant Geometry of Nilpotent Orbits and Associated Varieties

Peter CrooksDoctor of Philosophy

Graduate Department of MathematicsUniversity of Toronto

2016

Nilpotent orbits are highly structured algebraic varieties lying at the interface of alge-

braic geometry, Lie theory, symplectic geometry, and geometric representation theory.

The interest in these objects has been long-standing, ranging from Kostant’s founda-

tional work in the 1950s and 1960s to Kronheimer’s realization of nilpotent orbits as

instanton moduli spaces. At the same time, nilpotent orbits are studied for the sake of

understanding closely associated varieties, such as nilpotent Hessenberg varieties.

In this thesis, we study the equivariant algebraic geometry and topology of nilpo-

tent orbits and related varieties. Our first group of results is principally concerned

with presentations of T -equivariant cohomology rings. More specifically, we give con-

crete presentations of the T -equivariant cohomology rings of the regular and minimal

nilpotent orbits, with the latter presentation providing an equivariant counterpart to

existing work on the ordinary cohomology of the minimal nilpotent orbit. We also

examine the family of Hessenberg varieties arising from the minimal nilpotent orbit,

showing them to be GKM and obtaining presentations of their T -equivariant coho-

mology rings. In Lie type A, we explain how one would compute the Poincare poly-

nomials and irreducible components of these Hessenberg varieties.

Our second group of results includes a characterization of those semisimple real

Lie algebras for which every complex nilpotent orbit contains a real one, building on

Rothschild’s criterion for such a Lie algebra to be quasi-split. We also consider the role

of nilpotent orbits in quaternionic Kahler geometry by giving a new, self-contained

proof of the LeBrun-Salamon Conjecture for equivariant contact structures on partial

ii

flag varieties. This approach allows us to give an intrinsic description of the standard

contact structure on the isotropic Grassmannian of 2-planes in C2n.

iii

Contents

1 Introduction 1

1.1 Broad Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Objective and Overview of Results . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Preliminaries 6

2.1 Introduction and Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Lie-Theoretic Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3.2 Specialization to Type A . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4 Partial Flag Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4.1 T -Fixed Points and Schubert Varieties . . . . . . . . . . . . . . . . 9

2.4.2 Examples in Type A . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4.3 Equivariant Vector Bundles . . . . . . . . . . . . . . . . . . . . . . 11

2.5 Equivariant Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.5.1 The General Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.5.2 T -Equivariant Cohomology . . . . . . . . . . . . . . . . . . . . . . 15

2.6 GKM Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.7 GKM Theory for Partial Flag Varieties . . . . . . . . . . . . . . . . . . . . 20

2.7.1 Some General Considerations . . . . . . . . . . . . . . . . . . . . . 20

2.7.2 Example: The GKM Graph of a Full Flag Variety in Type A . . . . 21

2.8 Nilpotent Orbits in Semisimple Lie Algebras . . . . . . . . . . . . . . . . 22

2.8.1 Basic Definitions and Features . . . . . . . . . . . . . . . . . . . . 22

2.8.2 Nilpotent Orbits in Type A . . . . . . . . . . . . . . . . . . . . . . 27

iv

3 The T -Equivariant Cohomology of Nilpotent Orbits 283.1 Introduction and Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2 The Regular Nilpotent Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . 293.3 The Minimal Nilpotent Orbit . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3.1 The Equivariant Geometry of P(O) . . . . . . . . . . . . . . . . . . 313.3.2 The Moment Polytope of P(Omin) . . . . . . . . . . . . . . . . . . . 333.3.3 A Description of Omin and P(Omin) . . . . . . . . . . . . . . . . . . 333.3.4 The T -Equivariant Cohomology of Omin . . . . . . . . . . . . . . . 353.3.5 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4 Hessenberg Varieties for the Minimal Nilpotent Orbit 384.1 Introduction and Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2.1 The Hessenberg Varieties of Interest . . . . . . . . . . . . . . . . . 404.2.2 Examples in Type A . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.3 The Equivariant Geometry of XH(eλ) . . . . . . . . . . . . . . . . . . . . . 424.3.1 Algebraic Group Actions on XH(eλ) . . . . . . . . . . . . . . . . . 424.3.2 The Euler Number of XH(eλ) . . . . . . . . . . . . . . . . . . . . . 434.3.3 The Size of XH(eλ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.4 Poincare Polynomials and Irreducible Components . . . . . . . . . . . . 464.4.1 Poincare Polynomials in Type A . . . . . . . . . . . . . . . . . . . 464.4.2 Irreducible Components in Type ADE . . . . . . . . . . . . . . . . 494.4.3 Complete Description of the Irreducible Components in Type A . 52

4.5 GKM Theory on XH(eλ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.5.1 The GKM Graph of XH(eλ) . . . . . . . . . . . . . . . . . . . . . . . 564.5.2 GKM Graphs of XH(eλ) in Type A . . . . . . . . . . . . . . . . . . 58

4.6 Cohomology Ring Presentations . . . . . . . . . . . . . . . . . . . . . . . 604.6.1 Ordinary Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . 604.6.2 Equivariant Cohomology . . . . . . . . . . . . . . . . . . . . . . . 64

5 Nilpotent Orbit Complexification 685.1 Introduction and Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.2.1 Real and Complex Nilpotent Orbits . . . . . . . . . . . . . . . . . 695.2.2 The Closure Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.2.3 Partitions of Nilpotent Orbits . . . . . . . . . . . . . . . . . . . . . 71

5.3 Nilpotent Orbit Complexification . . . . . . . . . . . . . . . . . . . . . . . 73

v

5.3.1 The Complexification Map . . . . . . . . . . . . . . . . . . . . . . 735.3.2 Surjectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.3.3 The Image of ϕg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785.3.4 Fibres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6 Equivariant Contact Geometry and the LeBrun-Salamon Conjecture 826.1 Introduction and Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 826.2 Review of Properties of Fano Contact Varieties . . . . . . . . . . . . . . . 846.3 Partial Flag Varieties and Contact Structures . . . . . . . . . . . . . . . . 85

6.3.1 Basic Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.3.2 The Projectivization of the Minimal Nilpotent Orbit . . . . . . . . 866.3.3 Reduction to the Case of a Maximal Parabolic . . . . . . . . . . . 876.3.4 The Contact Line Bundle on G/PS . . . . . . . . . . . . . . . . . . 87

6.4 A Classification of G-Invariant Contact Structures on G/P . . . . . . . . . 896.4.1 The Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 896.4.2 Example: The Grassmannian of Isotropic 2-Planes in C2n . . . . . 90

Bibliography 93

vi

Chapter 1

Introduction

1.1 Broad Context

Conjugacy classes of matrices are concrete, well-studied objects that arise frequentlyin both classical and modern mathematics. In particular, there are many situationsin which one specializes to the conjugacy classes of nilpotent matrices. Such classesenjoy a number of properties that distinguish them from other families of conjugacyclasses. Indeed, over the complex numbers, Jordan canonical forms make it possibleto index the conjugacy classes of nilpotent matrices with partitions. As a consequence,there are only finitely many conjugacy classes of nilpotent matrices.

Everything mentioned above has a Lie-theoretic interpretation, in which conjugacyclasses of matrices generalize to adjoint orbits in a Lie algebra. Under this generaliza-tion, conjugacy classes of nilpotent matrices become so-called nilpotent orbits. Theselatter objects are studied at the interface of several sub-disciplines in mathematics.For instance, it is an ongoing algebro-geometric problem to describe the singularitiesarising in nilpotent orbit closures (see [43, 57]). In symplectic geometry, the Killingform establishes a correspondence between nilpotent orbits and conical (ie. dilation-invariant) coadjoint orbits. Combinatorialists have studied the natural poset structureon the collection of nilpotent orbits (see 2.51 for a definition of the partial order). Forgeometric representation theorists, nilpotent orbits arise in the context of Springer res-olution (see [17]). Finally, some work has been done to compute algebraic topologicalinvariants of nilpotent orbits (see [10, 18, 46]).

While they are independently interesting, nilpotent orbits give rise to other notablealgebraic varieties. Specifically, each (non-zero) nilpotent orbit can be projectivized toyield a subvariety of projective space. The orbit’s canonical symplectic structure then

1

CHAPTER 1. INTRODUCTION 2

descends to a complex contact structure on the projectivization. In fact, the long-standing LeBrun-Salamon Conjecture [62] posits that every Fano contact variety withsecond Betti number 1 arises as the projectivization of a so-called minimal nilpotentorbit.

Nilpotent orbits are also closely related to the so-called nilpotent Hessenberg va-rieties [23]. The latter are closed, singular subvarieties of the flag variety occurringin combinatorial algebraic geometry [65, 72, 73], geometric representation theory [33],and equivariant topology [3].

1.2 Objective and Overview of Results

The principal objective of this thesis is to systematically study the equivariant geom-etry of nilpotent orbits and associated varieties (ex. projectivizations and Hessenbergvarieties). We also include some related projects undertaken by the author during thedoctoral program. Organized by chapter, the main results of this thesis are as follows.

• In Chapter 3, we give explicit presentations of the T -equivariant cohomologyrings of the regular and minimal nilpotent orbits of a simple complex group (seeTheorem 7). We thereby obtain an equivariant counterpart to Juteau’s descrip-tion [46] of the ordinary cohomology of the minimal nilpotent orbit. This chapteris based on the paper [20].

• Chapter 4 introduces the class of Hessenberg varieties associated with the min-imal nilpotent orbit. We show them to be GKM varieties, and we give presen-tations of their ordinary and T -equivariant cohomology rings. In Lie type A,we develop an explicit combinatorial procedure for determining their Poincarepolynomials and irreducible components. This chapter is based on joint workwith Hiraku Abe, some of which appears in the preprint [2].

• In Chapter 5, we characterize those semisimple real Lie algebras g with the prop-erty that every nilpotent orbit in gC contains a nilpotent orbit in g (see Theorem46). This result builds on Rothschild’s criterion for g to be quasi-split (see Propo-sition 5.1 of [61]). This chapter is based on the preprint [19].

• Chapter 6 gives a modern, self-contained, Lie-theoretic proof of Boothby’s clas-sification of homogeneous complex contact varieties [11,12] in a special case (see


Theorem 60). Our approach yields an intrinsic description of the canonical con-tact structure on the isotropic Grassmannian of 2-planes in C2n. This chapter isbased on the joint work with Steven Rayan from [21].

Further details about the distribution of work in the joint projects can be found atthe beginning of the respective chapters.

1.3 Structure of the Thesis

We now outline this thesis as a whole. Noting that each chapter begins with a sum-mary of its contents, our outline will be broad in scope.

Chapter 2 is devoted to some of the notation, conventions, and background ma-terial one requires for subsequent chapters. This material includes, but is not limitedto, partial flag varieties, equivariant cohomology, GKM theory, and nilpotent orbits.Chapter 3 then calculates the T -equivariant cohomology rings of the regular and min-imal nilpotent orbits. The second calculation is the more complicated, and dependsfundamentally on GKM theory and nilpotent orbit projectivizations.

Chapter 4 defines and studies the family of Hessenberg varieties associated withthe minimal nilpotent orbit. While we establish several properties of these varieties,emphasis is placed on two main themes. Firstly, we use combinatorial and diagram-matic gadgets to find the Poincare polynomials and irreducible components of thesevarieties in Lie Type A. Secondly, for each variety, we obtain two independent pre-sentations of the T -equivariant cohomology ring. One of these is a consequence ofshowing our variety to be GKM, while the other comes from exhibiting the equivari-ant cohomology ring as a quotient of H∗T(G/B).

Chapter 5 explores relationships between real and complex nilpotent orbits bymeans of nilpotent orbit complexification. Using this framework, we classify the semisim-ple real Lie algebras g with the property that every nilpotent orbit in gC meets g. Anyg satisfying this property is necessarily quasi-split, and for such g we characterize thecomplex nilpotent orbits meeting g. The chapter concludes with the result that distinctreal nilpotent orbits lying in the same complex orbit are incomparable in the closureorder. Our proof invokes the Kostant-Sekiguchi Correspondence and its properties.

In Chapter 6, our focus turns to the nilpotent orbits in quaternionic Kahler geom-etry. Specifically, we give a proof of the LeBrun-Salamon Conjecture for partial flagvarieties with G-invariant complex contact structures. While this result is deduciblefrom Boothby’s work [11,12], our proof instead harnesses the equivariant geometry of


partial flag varieties. Also, our approach is shown to imply an intrinsic description ofthe standard contact structure on the isotropic Grassmannian of 2-planes in C2n.


1.4 Acknowledgements

Let me begin by acknowledging Lisa Jeffrey and John Scherk for their joint supervisionof this doctoral thesis. Their remarkable support, insight, and kindness warrant myeternal gratitude. I would also like to thank Marco Gualtieri, Joel Kamnitzer, YaelKarshon, Eckhard Meinrenken, and Paul Selick for enhancing and contributing to myresearch experience at the University of Toronto.

I am grateful for the financial support provided by the Natural Sciences and Engi-neering Research Council of Canada and the Ontario Ministry of Training, Colleges,and Universities.

Throughout the doctoral program, I have benefited from extensive conversationswith collaborators and colleagues. In this context, I would like to acknowledge HirakuAbe, Faisal Al-Faisal, Alexander Caviedes Castro, Payman Eskandari, Jonathan Fisher,Iva Halacheva, Tyler Holden, Kevin Luk, Ali Mousavidehshikh, Steven Rayan, andDaniel Rowe. I would also like to recognize Alfonso Gracia-Saz and Sean Uppal forthe fundamental dignity and professionalism they bring to teaching in mathematics.They have helped to shape my approach to course instruction and coordination.

I would like to acknowledge the members of our department’s administrative stafffor their competence, kindness, and optimism. This description is particularly fittingin the case of Ida Bulat, whose incomparable presence is greatly missed.

Chapter 2

Preliminaries

2.1 Introduction and Structure

This chapter assembles some of the notation, conventions, and background on whichsubsequent chapters will depend. We begin by listing some of the fundamental con-ventions observed throughout this thesis. Section 2.3 subsequently introduces themain Lie-theoretic objects used in Chapters 3, 4, and 6. Next, Section 2.5 recalls the ba-sics of equivariant cohomology. Using this as a foundation, Sections 2.6 and 2.7 reviewGKM theory and explain how it applies to compute the T -equivariant cohomology ofa partial flag variety. Section 2.8 provides some of the basic background on nilpotentorbits in semisimple Lie algebras.

2.2 Conventions

In the interest of consistency, let us establish some of the fundamental notation andconventions to be used throughout this thesis.

• We will make extensive use of algebraic and geometric concepts for which, strictlyspeaking, a base field must be specified (ex. Lie algebras, algebraic varieties, etc.)However, parsimony is best served by our sometimes not mentioning the basefield. In all such cases (except for the one mentioned in the next bullet point),this field is to be taken as C.

• Whenever we invoke homology and cohomology (ordinary, equivariant, etc.)without specifying a coefficient ring, let us take this ring to be Q.

6

CHAPTER 2. PRELIMINARIES 7

• Whenever we introduce a group action without indicating it to be a left or rightaction, we will understand it to be a left action.

• Whenever we refer to a topological invariant of an algebraic variety (ex. ordinarycohomology), we will do so with the understanding that the variety possessesits classical topology. Similarly, all equivariant topological invariants (ex. equiv-ariant cohomology) will be computed with respect to the classical topologies onboth the varieties and groups in question.

• Throughout Chapters 2, 3, 4, and 6, the groups G, B, B−, and T will be exactly asdefined in 2.3.1.

• By vector bundle, we will mean an algebraic vector bundle E on a smooth varietyX. If x ∈ X, then Ex will denote the fibre over x.

2.3 Lie-Theoretic Objects

2.3.1 Notation

Let G be a connected, simply-connected simple algebraic group with Lie algebra g.Denote by

Ad : G→ GL(g) and ad : g → gl(g)

the adjoint representations of G and g, respectively. Fix a pair of positive and negativeBorel subgroups, B ⊆ G and B− ⊆ G, respectively, and consider the maximal torusT := B ∩ B−. The inclusions T ⊆ B ⊆ G give rise to inclusions of Lie algebras, whichwe denote by t ⊆ b ⊆ g. One then has the weight lattice X∗(T), roots ∆ ⊆ X∗(T),positive roots ∆+ ⊆ ∆, negative roots ∆− ⊆ ∆, and simple roots Π ⊆ ∆+. The weightlattice comes equipped with its usual bilinear form 〈·, ·〉 : X∗(T) ⊗Z X∗(T) → Z. Also,since G is simple, there is a highest root λ ∈ ∆+. Finally, let W = NG(T)/T denote theWeyl group, sγ ∈ W the reflection associated to γ ∈ ∆+, and ` : W → Z≥0 the lengthfunction resulting from our choice of simple roots.

Recall that a parabolic subgroup P ⊆ G is called standard if B ⊆ P. Assuming this tobe the case, set S := α ∈ Π : g−α ⊆ p, where p is the Lie algebra of P. The assignmentof S to P defines a bijective correspondence between the standard parabolic subgroupsand the subsets of Π. Given S ⊆ Π, we shall let PS ⊆ G denote the standard parabolicsubgroup associated to S via this correspondence, and we shall let pS ⊆ g denote its


Lie algebra. LetWS denote the Weyl group of PS, and recall thatWS is the subgroup ofW generated by the simple reflections sα, α ∈ S.

2.3.2 Specialization to Type A

While we will usually work in arbitrary Lie type, we will sometimes benefit fromspecializing to type An−1. In these cases only, the notation in 2.3.1 is to be interpretedas follows.

• G = SLn(C) = g ∈ GLn(C) : det(g) = 1

• g = sln(C) = X ∈ gln(C) : tr(X) = 0

• For g ∈ SLn(C) and X ∈ sln(C), we have

Ad(g)(X) = gXg−1. (2.1)

• B and B− are the subgroups of upper and lower-triangular matrices in SLn(C),respectively.

• T is the subgroup of diagonal matrices in SLn(C), namely

T =

t1 0 0 . . . 0

0 t2 0 . . . 0...

...... . . . ...

0 0 0 . . . tn

: t1t2 · · · tn = 1

. (2.2)

• Abusing notation, ti ∈ X∗(T) shall denote the weight

T → C∗,

t1 0 0 . . . 0

0 t2 0 . . . 0...

...... . . . ...

0 0 0 . . . tn

7→ ti.

Note that X∗(T) is then generated by t1, t2, . . . , tn with the relationt1 + t2 + . . .+ tn = 0.

• ∆ = ti − tj : 1 ≤ i, j ≤ n, i 6= j∆+ = ti − tj : 1 ≤ i < j ≤ n


Π = ti − ti+1 : 1 ≤ i ≤ n− 1

λ = t1 − tn.

• We will regard W as the symmetric group Sn, so that the Weyl-action on X∗(T)satisfies

wti = tw(i), w ∈ Sn, i = 1, 2, . . . , n. (2.3)

• We will sometimes use the so-called one-line notation for permutations, in whichone represents w ∈ Sn by the list of values w(1) w(2) · · · w(n).

• If 1 ≤ i < j ≤ n, then (i j) ∈ Sn will denote the transposition switching i and j.Note that (i j) is precisely sγ for γ = ti − tj.

2.4 Partial Flag Varieties

The following section reviews some relevant aspects of partial flag varieties, objectsof central importance to this thesis. To this end, recall that a partial flag variety (of G)is a projective variety equipped with a transitive algebraic G-action. Equivalently, apartial flag variety is a projective G-variety with the property of being equivariantlyisomorphic to G/PS for some subset S ⊆ Π. We shall refer to G/B, the partial flagvariety with maximal dimension, as the full flag variety.

2.4.1 T -Fixed Points and Schubert Varieties

Recall that T acts on G/PS with finitely many fixed points, indexed by W/WS in thefollowing way. Given [w] ∈ W/WS, lift [w] to w ∈ W, and then lift w to g ∈ NG(T).Let x([w]) ∈ G/PS denote the coset of g in G/PS. This point is defined independentlyof our earlier choices, and the association

W/WS → (G/PS)T , [w] 7→ x([w]) (2.4)

is a bijection.

The groups B and B− act on G/PS with respective orbit decompositions

G/PS =∐

[w]∈W/WS

Bx([w]) (2.5)


andG/PS =

∐[w]∈W/WS

B−x([w]). (2.6)

The orbit Bx([w]) is an affine space of dimension `([w]), where `([w]) denotes thelength of a coset representative of [w] having minimal length. Similarly, B−x([w]) isan affine space of codimension `([w]) in G/PS. One calls Bx([w]) (resp. B−x([w])) aSchubert cell (resp. an opposite Schubert cell), alluding to the fact that (2.5) and (2.6) arecell decompositions of G/PS.

The closures of Schubert and opposite Schubert cells give a geometric manifesta-tion of the Bruhat order (see [25]) onW/WS in the sense that

Bx([w]) =∐

[v]≤[w]

Bx([v]) (2.7)

andB−x([w]) =

∐[w]≤[v]

B−x([v]). (2.8)

The subvariety Bx([w]) is called a Schubert variety, while B−x([w]) is called an op-posite Schubert variety. Henceforth, we shall denote the former by X([w]) and the latterby X−([w]).

It will be important to note that the collections

σ([w]) := [X([w])] ∈ H∗(G/PS;Z), [w] ∈W/WS (2.9)

andσ−([w]) := [X−([w])] ∈ H2`([w])(G/PS;Z), [w] ∈W/WS (2.10)

are additive bases of H∗(G/PS;Z). The classes in (2.9) are called Schubert classes, whilethose in (2.10) are called opposite Schubert classes.

2.4.2 Examples in Type A

The partial flag varieties of SLn(C) are realizable in terms of flags of subspaces of Cn.Indeed, for integers 1 ≤ d1 < d2 < . . . < dk ≤ n − 1, denote by F(d1, d2, . . . , dk;Cn)the set of all flags V• of the following form:

V• = (0 ⊆ V1 ⊆ V2 ⊆ . . . ⊆ Vk ⊆ Cn),


where Vj is a dj-dimensional subspace ofCn. This set has a canonical projective varietystructure, and the natural SLn(C)-action then renders it a partial flag variety.

We note that the full flag variety is the one for which flags have maximal length,namely

SLn(C)/B ∼= F(1, 2, . . . , n− 1;Cn). (2.11)

Equivalently, letting Ek := spane1, e2, . . . , ek be the span of the first k standard basisvectors, one notices that B is precisely the SLn(C)-stabilizer of the flag

E• := (0 ⊆ E1 ⊆ E2 ⊆ . . . ⊆ En−1 ⊆ Cn) ∈ F(1, 2, . . . , n− 1;Cn).

In the interest of concreteness, let us outline how our earlier discussion of T -fixedpoints and Schubert varieties applies to F(1, 2, . . . , n − 1;Cn). Indeed, noting thatB = P∅ and W∅ = e, (2.4) implies that the T -fixed points are indexed by the fullWeyl group W = Sn. Now, given w ∈ Sn and k ∈ 1, 2, . . . , n − 1, set Ewk :=

spanew(1), ew(2), . . . , ew(k) and

Ew• := (0 ⊆ Ew1 ⊆ Ew2 ⊆ . . . ⊆ Ewn−1 ⊆ Cn) ∈ F(1, 2, . . . , n− 1;Cn).

The T -fixed points and Schubert varieties in F(1, 2, . . . , n− 1;Cn) are then given by

F(1, 2, . . . , n− 1;Cn)T = Ew• : w ∈ Sn

and

X(w) = BEw• =

V• ∈ F(1, 2, . . . , n− 1;Cn) :dim(Vj ∩ Ek) ≥ dim(Vj ∩ Ewk )

1 ≤ j ≤ n− 1

1 ≤ k ≤ n− 1

,respectively. For a more extensive discussion of these descriptions, we refer the readerfo [31].

2.4.3 Equivariant Vector Bundles

Let G be an algebraic group and X a variety on which G acts algebraically. Recall thata vector bundle E on X is called G-equivariant if

• G acts algebraically on E via a lift of the G-action on X, and

• for all x ∈ X and g ∈ G, the action of g on E restricts to a vector space isomor-


phism Ex∼=−→ Egx.

Given a G = G-equivariant vector bundle E on X = G/PS, note that E[e] is a PS-representation (where [e] ∈ G/PS denotes the coset of the identity e ∈ G). Conversely,we may associate to each finite-dimensional PS-representation ϕ : PS → GL(V) a G-equivariant vector bundle over G/PS. Explicitly, note that PS acts freely on G× V via

p · (g, v) := (gp−1, ϕ(p)(v)), p ∈ PS, g ∈ G, v ∈ V. (2.12)

The quotient variety, denoted G ×PS V , carries a residual G-action and has a naturalmap to G/PS whose fibres are vector spaces. In this way, G ×PS V is a G-equivariantvector bundle over G/PS.

With E and V as above, one has

G×PS E[e]∼= E (2.13)

as G-equivariant vector bundles and

(G×PS V)[e] ∼= V (2.14)

as PS-representations. In other words, the associations E 7→ E[e] and V 7→ G ×PS Vare inverses in a bijective correspondence between the isomorphism classes of G-equivariant vector bundles onG/PS and those of finite-dimensional PS-representations.

Our correspondence restricts to give one between Hom(PS,C∗) and the (isomor-phism classes of) G-equivariant line bundles on G/PS. This gives rise to an injection

Hom(PS,C∗) → Pic(G/PS), (2.15)

where Pic(G/PS) denotes the Picard group ofG/PS. Noting that restriction to T definesan isomorphism Hom(PS,C∗)

∼=−→ X∗(T)WS , we may present (2.15) as

X∗(T)WS → Pic(G/PS). (2.16)

Now for each α ∈ X∗(T)WS , consider

L(α) := G×PS Cα, (2.17)

where Cα is the one-dimensional PS-representation of T -weight α. The map (2.16) is


then given byX∗(T)WS → Pic(G/PS), α 7→ [L(α)]. (2.18)

Proposition 1. The map (2.18) is a group isomorphism.

Proof. It is known that the structure sheaf of G/PS has trivial first and second coho-mology groups (see [66]). An examination of the exponential sequence in sheaf coho-mology then implies that

Pic(G/PS) → H2(G/P;Z), [L] 7→ c1(L) (2.19)

is an isomorphism. Hence, it will suffice to prove that the composition of (2.18) and(2.19) is an isomorphism. This composite map is given by

X∗(T)WS → H2(G/PS;Z), α 7→ c1(L(α)), (2.20)

which is known to be an isomorphism (see [8]).

2.5 Equivariant Cohomology

2.5.1 The General Setup

Equivariant cohomology will be our principal means of describing objects in the equiv-ariant topological category. Here, we give a narrowly focused treatment of equivari-ant cohomology, emphasizing only those aspects directly pertinent to this thesis. Fora more exhaustive study, we refer the reader to [14].

For G a topological group, there exists a contractible topological space EG on whichG acts continuously and freely. The resulting principal G-bundle

EG → EG/G =: BG (2.21)

is called a universal principal G-bundle. This nomenclature reflects the universal prop-erty that every principal G-bundle E→ Y arises as a pullback of (2.21) along a suitablemap Y → BG (defined uniquely up to homotopy).

Given a topological space X carrying a continuous G-action, let G act diagonally onthe product X× EG. The quotient

(X× EG)/G =: XG (2.22)


is called the Borel mixing space of X. The G-equivariant cohomology of X is then definedby

H∗G(X) := H∗(XG). (2.23)

Using the universal property of (2.21), one can show this Z-graded commutative Q-algebra to be well-defined up to isomorphism.

It will be prudent to mention a few elementary properties of G-equivariant co-homology. To this end, suppose that X and Y have continuous G-actions, and thatf : X→ Y is a G-equivariant continuous map. Note that f induces a continuous map

fG : XG → YG, (2.24)

which in turn gives a map on cohomology

f∗G : H∗G(Y) → H∗G(X). (2.25)

Let us consider the following special cases of this construction.

• Module Structure: The map from X to the one-point space, X → pt, induces amap of rings, H∗G(pt) → H∗G(X). In this way, H∗G(X) is canonically a module overH∗G(pt).

• Restriction: The inclusion XG → X is G-equivariant and therefore induces a mapH∗G(X) → H∗G(X

G). We will often refer to this latter map as a restriction map.

• Free Actions: Let G act freely on X and consider the map

(X× EG)/G → X/G

defined by projection from the first component. This is a fibration with con-tractible fibre EG, and it therefore induces an isomorphism on cohomology. Sincethe cohomology of the total space is precisely H∗G(X), we have

H∗G(X)∼= H∗(X/G).

• Homogeneous Spaces: Let K ⊆ G be a closed subgroup and consider the homo-geneous G-space X := G/K. The map

EG/K → ((G/K)× EG)/G, [y] 7→ [([e], y)] (2.26)


is seen to be a well-defined homeomorphism, where [e] ∈ G/K is the coset of theidentity e ∈ G. In particular, the domain and codomain of (2.26) have isomorphiccohomology rings. The cohomology of the latter is exactly H∗G(G/K). Also, as EGis a contractible space on which K acts freely, we may set BK := EG/K. Hence,the cohomology of the codomain is H∗(BK) = H∗K(pt) and we have

H∗G(G/K) ∼= H∗K(pt). (2.27)

• Group Complexification: Let K be a compact connected Lie group with com-plexification KC. Given a variety X with an algebraic KC-action, one has a fibra-tion

KC/K→ (X× EKC)/K→ (X× EKC)/KC. (2.28)

The fibre KC/K is contractible and (2.28) induces an isomorphism on ordinarycohomology,

H∗((X× EKC)/KC)∼=−→ H∗(X× EKC)/K). (2.29)

Note that the left-hand-side of (2.29) is precisely H∗KC(X). Also, as EKC is a con-

tractible space on which K acts freely, we may take (X × EKC)/K to be the Borelmixing space of X with respect to the K-action. In this way, the right-hand-sideof (2.29) is H∗K(X), giving us a canonical isomorphism

H∗KC(X)

∼=−→ H∗K(X). (2.30)

• Equivariant Chern Classes: If π : E → X is a G-equivariant vector bundle (asdefined in 2.4.3), then πG : EG → XG is naturally a vector bundle. This fact givesrise to G-equivariant Chern classes cGn(E), namely

cGn(E) := cn(EG) ∈ H2n(XG) = H2nG (X). (2.31)

2.5.2 T -Equivariant Cohomology

We will be principally concerned with T -equivariant cohomology, to which the theoryin 2.5.1 has interesting specializations.

• The T -Equivariant Cohomology of a Point: For α ∈ C∗, let Cα denote the one-dimensional T -representation of weight α. Note that Cα is equivalently a T -equivariant line bundle over pt, and as such has T -equivariant first Chern class


cT1(Cα) ∈ H2T(pt). The association α 7→ cT1(Cα) determines a degree-doublingQ-algebra isomorphism

Sym(X∗(T)⊗Z Q)∼=−→ H∗T(pt). (2.32)

Using (2.32), we will freely identify H∗T(pt) and Sym(X∗(T)⊗Z Q).

Note that ifΠ = α1, α2, . . . , αr, then Sym(X∗(T)⊗ZQ) is more concretely viewedas Q[α1, α2, . . . , αr]. Hence, we will occasionally identify H∗T(pt) with this poly-nomial algebra.

• Equivariant Formality: Let X be a variety on which T acts algebraically. Onecalls X equivariantly formal if the cohomology spectral sequence of the naturalfibration

Xι−→ XT → BT (2.33)

collapses on its second page, so that

H∗T(X)∼= H∗(X)⊗Q H∗T(pt)

as H∗T(pt)-modules. The class of equivariantly formal spaces includes, for in-stance, all T -varieties with vanishing odd-degree cohomology.

If X is equivariantly formal, then the map ι from (2.33) induces a surjection ι∗ :H∗T(X) → H∗(X). Its kernel is H>0T (pt)H∗T(X), where H>0T (pt) is the ideal of H∗T(pt)generated by the positive-degree elements. In particular,H∗(X) can be recoveredfrom H∗T(X) via the ring isomorphism

H∗(X) ∼=H∗T(X)

H>0T (pt)H∗T(X).

• T -Equivariant Schubert Classes: It will be advantageous to note that the coho-mology classes (2.9) and (2.10) have counterparts in T -equivariant cohomology.Given [w] ∈W/WS, let

σT([w]) ∈ H∗T(G/PS) (2.34)

andσ−T ([w]) ∈ H

2`([w])T (G/PS) (2.35)

denote the T -equivariant cohomology classes determined by X([w]) and X−([w]),respectively. The former is called a T -equivariant Schubert class, while the latter is


called a T -equivariant opposite Schubert class.

For details on the association of equivariant cohomology classes to T -invariantsubvarieties, we refer the reader to Section 3.5 of [1].

2.6 GKM Theory

Having recalled the general theory of equivariant cohomology, let us discuss sometechniques for computation. To this end, we begin with arguably the most powerfuland celebrated such technique — one developed by Goresky, Kottwitz, and MacPher-son (henceforth abbreviated GKM) in [35]. These authors developed a concrete frame-work for computing the T -equivariant cohomology rings of a large and encompassingclass of varieties, the so-called GKM varieties.

Definition 2. A GKM variety is a projective variety X on which T acts algebraicallyand with the following properties:

(i) XT is finite.

(ii) X has finitely many one-dimensional T -orbits.

(iii) X is equivariantly formal.

Prominent examples of varieties satisfying Definition 2 include the partial flag vari-etiesG/PS (see 2.7), the Schubert varietiesX([w]), and smooth projective toric varieties.However, there have been some interest in broadening the class of spaces originallyconsidered by GKM. In [37], Guillemin and Holm relax the requirement that XT con-sist of isolated points. Also, Harada, Henriques, and Holm [39] substantially weakenthe assumptions in Definition 2 to compute the equivariant cohomology rings of Kac-Moody flag varieties.

Let us now discuss some implications of Definition 2 for a GKM variety X. Ac-cordingly, let Y ⊆ X be the closure of a one-dimensional T -orbit in X. There exists anon-zero weight γ ∈ X∗(T) for which Y is T -equivariantly isomorphic to P1 with theT -action

t · [x1 : x2] = [γ(t)x1 : x2]. (2.36)

Henceforth, we shall denote this latter T -variety by (P1, γ).Interestingly, the property Y ∼= (P1, γ) determines γ uniquely up to sign. This fact

will feature prominently in our discussion of GKM graph edges, and we prove it viathe following lemma.


Lemma 3. If γ1, γ2 ∈ X∗(T) are non-zero weights, then (P1, γ1) and (P1, γ2) are T -equivariantlyisomorphic if and only if γ1 = γ2 or γ1 = −γ2.

Proof. Assume that (P1, γ1) and (P1, γ2) are T -equivariantly isomorphic, and let ϕ :

(P1, γ1) → (P1, γ2) be an isomorphism. Since [1 : 0] and [0 : 1] are the T -fixed points ineach variety, we must have ϕ([0 : 1]) = [0 : 1] or ϕ([0 : 1]) = [1 : 0]. In the former case,the differential of ϕ at [0 : 1] gives an isomorphism

d[0:1]ϕ : T[0:1]P1∼=−→ T[0:1]P1 (2.37)

of one-dimensional T -representations.

Now, let U denote the complement of [1 : 0] in (P1, γ1), and let Cγ1 denote theone-dimensional T -representation of weight γ1. The map

Cγ1 → U, x0 7→ [x0 : 1]

is a T -equivariant isomorphism, and so induces a T -representation isomorphism

T[0:1]P1 ∼= T0Cγ1 ∼= Cγ1 .

In other words, the domain of (2.37) is acted upon by T with weight γ1. Similarly, theweight of the codomain is γ2, and we have γ1 = γ2.

In the case ϕ([0 : 1]) = [1 : 0], the differential of ϕ at [0 : 1] gives an isomorphism

d[0:1]ϕ : T[0:1]P1∼=−→ T[1:0]P1.

The weight of the domain is again γ1, while consideration of the complement of [0 : 1]in (P1, γ2) shows the weight of the codomain to be −γ2. Hence, γ1 = −γ2.

To prove the converse, assume that γ1 = γ2 or γ1 = −γ2. Since the desired conclu-sion clearly holds in the former case, we shall assume that γ1 = −γ2. Now, considerthe variety isomorphism

ψ : P1 → P1, [x0 : x1] 7→ [x1 : x0].


If t ∈ T and [x0 : x1] ∈ P1, then

ψ([γ1(t)x0 : x1]) = [x1 : γ1(t)x0]

= [γ1(t)−1x1 : x0]

= [(−γ1)(t)x1 : x0]

= [γ2(t)x1 : x0].

So, in addition to being a variety isomorphism, ψ is a T -equivariant map (P1, γ1) →(P1, γ2). This completes the proof.

Returning to the main discussion, let Y ⊆ X be the closure of a one-dimensionalT -orbit and let γ ∈ X∗(T) be a non-zero weight for which Y ∼= (P1, γ). Since (P1)T =

[1 : 0], [0 : 1], it follows that YT consists of two distinct T -fixed points in X. In thisway, certain pairs of points in XT are “connected” by the closures of one-dimensionalT -orbits. Given distinct x1, x2 ∈ XT , let us write x1 → x2 when these points lie in theclosure of a one-dimensional T -orbit. In this case, the closure Yx1,x2 is unique and actedupon by T with some non-zero weight γx1,x2 ∈ X∗(T), as in (2.36).

Perhaps surprisingly, the data of XT and the weights γx1,x2 for x1 → x2 completelydetermine H∗T(X) as a subalgebra of H∗T(X

T).

Theorem 4 (The GKM Presentation, [35]). Suppose that all objects are as defined above.The restriction map

H∗T(X) → H∗T(XT) =

⊕x∈XT

H∗T(x) =⊕x∈XT

H∗T(pt)

is injective, and its image is precisely

H∗T(X)∼= (fx)x∈XT ∈

⊕x∈XT

H∗T(pt) : γx1,x2 |(fx1 − fx2) whenever x1 → x2. (2.38)

Remark. For purposes of interpreting (2.38), fx ∈ H∗T(pt) is to be regarded as an elementof Sym(X∗(T)⊗ZQ). The condition γx1,x2 |(fx1− fx2) then makes sense in the symmetricalgebra.

The divisibility conditions appearing in (2.38) are conveniently encoded in an edge-labelled graph GKM(X), called the GKM graph of X. This graph has vertex set XT , withx1 6= x2 ∈ XT connected by an edge if and only if x1 → x2 as defined above. In thiscase, the edge in question is given the label γx1,x2 . While this label is defined only upto a choice of sign, the choice does not affect the right-hand-side of (2.38).


2.7 GKM Theory for Partial Flag Varieties

2.7.1 Some General Considerations

Equipped with their usual T -actions, the partial flag varieties of G are perhaps themost recognized examples of GKM varieties. In the interest of later sections, we nowreview the construction of the GKM graph of a partial flag variety G/PS. For this, wewill need to recall a classification of the closures of one-dimensional T -orbits in G/PS(as presented in [32], for instance).

Given γ ∈ ∆+, let SL2(C)γ ⊆ G denote the root subgroup with Lie algebra g−γ ⊕[g−γ, gγ]⊕ gγ. Consider the SL2(C)γ-orbit of [e] ∈ G/PS, namely

Yγ := SL2(C)γ[e] ⊆ G/PS. (2.39)

For an element [w] ∈W/WS, we set

Y[w],γ := gYγ, (2.40)

where g ∈ NG(T) is a lift of w ∈ W. Note that Y[w],γ is a non-singleton if and only ifγ 6∈ ∆S,+, the set of positive roots in the Z-span of S. In this case, Y[w],γ is the closure ofa one-dimensional T -orbit and

(Y[w],γ)T = x([w]), x([wsγ]). (2.41)

Proposition 5. If [w] ∈W/WS and γ ∈ ∆+ \∆S,+, then Y[w],γ is T -equivariantly isomorphicto (P1, wγ).

Proof. Firstly, note that the SL2(C)γ-stabilizer of [e] ∈ G/PS is

Bγ := B ∩ SL2(C)γ, (2.42)

a Borel subgroup of SL2(C)γ. Hence, we have a T -equivariant isomorphism

Yγ ∼= SL2(C)γ/Bγ. (2.43)

The latter variety is seen to equivariantly identify with (P1, γ), so that Yγ ∼= (P1, γ). Inother words, there exists a variety isomorphism ϕ : Yγ → P1 satisfying

ϕ(tx) = [γ(t)x0 : x1] (2.44)


for all t ∈ T and x ∈ Yγ, where ϕ(x) := [x0 : x1].Now, let g ∈ NG(T) represent w and recall that Y[w],γ = gYγ. With this in mind,

consider the variety isomorphism ψ : Y[w],γ → P1 defined by ψ(gx) = ϕ(x) for x ∈ Yγ.To prove the proposition, it will suffice to establish that

ψ(t(gx)) = [(wγ)(t)x0 : x1] (2.45)

for all t ∈ T and x ∈ Yγ (where, again, ψ(gx) = ϕ(x) = [x0 : x1]).We have

ψ(t(gx)) = ϕ((g−1tg)x)

= [γ(g−1tg)x0 : x1] (by (2.44))

= [(wγ)(t)x0 : x1],

completing the verification of (2.45).

It is known that the subvarieties Y[w],γ, [w] ∈ W/WS, γ ∈ ∆+ \ ∆S,+, constitute acomplete list of the closures of one-dimensional T -orbits in G/PS. In particular, wehave the means to concretely describe the graph GKM(G/PS).

• Using the description of (G/PS)T given in (2.4), we will regard GKM(G/PS) ashaving vertex setW/WS.

• By (2.41), two vertices are connected by an edge if and only if they are of theform [w] and [wsγ] for w ∈W and γ ∈ ∆+ \ ∆S,+. Noting Proposition 5, the edgein question is labelled with wγ.

Recalling Theorem 4, our description of GKM(G/PS) is equivalent to the followingGKM presentation of H∗T(G/PS).

H∗T(G/PS)∼=

(f[w]) ∈⊕

[w]∈W/WS

H∗T(pt) :(wγ)|(f[w] − f[wsγ])

∀w ∈W∀γ ∈ ∆+ \ ∆S,+

(2.46)

2.7.2 Example: The GKM Graph of a Full Flag Variety in Type A

Let us draw the GKM graph of F(1, 2;C3), the full flag variety of SL3(C). To this end,note that F(1, 2;C3) = SL3(C)/B = SL3(C)/PS for S = ∅. In particular, WS is thetrivial group and ∆S,+ = ∅. With these observations in mind, the framework from 2.7.1applies to give the following description of GKM(F(1, 2;C3)).


• The vertex set coincides with the Weyl group, which in this case is S3. We willrecord these vertices using the one-line notation for elements of S3.

• Two vertices w,w ′ ∈ S3 are joined by an edge if and only if w ′ = w(i j) for some1 ≤ i < j ≤ 3. In this case, the edge is given the label w(ti − tj) = tw(i) − tw(j).

Setting γij := ti − tj, the following is the GKM graph itself.

t

t

t tt t

HHHHHaaaaaaaaaa

!!!!!!!!!!HH

HHH

1 2 3

3 2 1

2 3 1

1 3 2

3 1 2

2 1 3 γ12 γ23

γ13 γ13

γ23 γ12

γ23 γ12

γ13

Figure 2.1: The GKM graph of F(1, 2;C3)

2.8 Nilpotent Orbits in Semisimple Lie Algebras

The following section provides some of the relevant background and context for ourstudy of nilpotent orbits in semisimple Lie algebras.

2.8.1 Basic Definitions and Features

Recall that ξ ∈ g is called nilpotent if ad(ξ) : g → g is nilpotent as a vector space en-domorphism. More powerfully, if one fixes any faithful finite-dimensional g-moduleφ : g → gl(V), then ξ is nilpotent if and only if φ(ξ) : V → V is a nilpotent endo-morphism (see [13], Chapter I). In particular, if g is presented as a Lie subalgebra ofglN(C), then the nilpotent elements of g are precisely the nilpotent matrices lying in g.

The locus of all nilpotent ξ ∈ g is called the nilpotent cone N ⊆ g, namely

N := ξ ∈ g : ξ is nilpotent. (2.47)

This closed subvariety of g is known to be irreducible and singular (see (2.49) for moredetails on the latter property). More immediately, however, N is invariant under theadjoint representation of G. Equivalently, N is a union of adjoint G-orbits.


Definition 6. An adjoint G-orbit in N is called a nilpotent orbit.

We now mention a few of the salient properties of nilpotent orbits.

• Locally Closed: Being algebraic group orbits, nilpotent orbits are smooth, locallyclosed subvarieties of N. However, only the trivial orbit 0 is closed in N.

• Finiteness: Applying the work of Kostant and others, one sees that there areonly finitely many nilpotent orbits. Indeed, consider the inclusion C[g]G → C[g]and the induced map of varieties

χ : g → Spec(C[g]G). (2.48)

The variety Spec(C[g]G) identifies with affine space in such a way that N co-incides with the fibre of χ over 0. The finiteness of nilpotent orbits then fol-lows from Kostant’s result that each fibre of χ is a finite union of adjoint orbits(see [53]).

• Symplectic Structures: Each nilpotent orbit O ⊆ N is canonically a symplecticvariety. Indeed, the Killing form is non-degenerate and induces an isomorphismbetween the adjoint and coadjoint representations. Through this isomorphism,O corresponds to a coadjoint orbit with its Kirillov-Kostant-Souriau symplecticstructure (see [7], Section 7.5).

• Instanton Moduli Spaces: Nilpotent orbits arise as “instanton moduli spaces”in theoretical physics. More precisely let K ⊆ G be a compact real form withLie algebra k ⊆ g. One considers smooth curves A : R → Hom(su(2), k) intothe vector space of R-linear maps su(2) → k. The space of such curves carriesa natural action of K, and we shall let C(ρ) denote the K-orbit of a Lie algebramorphism ρ : su(2) → k. The instanton moduli space M(ρ) then consists of thosesmooth curves A : R→ Hom(su(2), k) satisfying

limt→∞A(t) ∈ C(ρ),

as well as the anti-self-dual Yang-Mills equations on SU(2)× R (see [28]).

Now, ρ complexifies to give ρC : sl2(C) → g. It is a celebrated result of Kron-heimer’s work [55] that M(ρ) is canonically diffeomorphic to the nilpotent G-


orbit through ρC(e), where

e =

[0 1

0 0

]∈ sl2(C)

is the standard nilpositive vector.

Conversely, suppose that O ⊆ g is a nilpotent orbit. Given any point ξ ∈ O, theJacobson-Morozov Theorem allows one to find a complex Lie algebra map φ :

sl2(C) → g such that ξ = φ(e). By replacing φ with a G-conjugate if necessary,we may assume that φ(su(2)) ⊆ k. It follows that φ = ρC, where ρ : su(2) → k

is the restriction of φ to su(2). Kronheimer’s result then identifies O with M(ρ).Hence, our discussion demonstrates that every nilpotent G-orbit arises as aninstanton moduli space.

• Springer Theory: Nilpotent orbits are ubiquitous in Springer Theory (see [17]).In more detail, one has the so-called Springer resolution

µ : G×B n → N, [(g, ξ)] 7→ Ad(g)(ξ), (2.49)

where n :=⊕

γ∈∆+gγ. The fibres of µ are called Springer fibres. Notably, the

isomorphism class of µ−1(ξ) depends only on the nilpotent orbit containing ξ.

• The Dilation Action: Nilpotent orbits are distinguished from general adjointorbits in that only the former are dilation-invariant. Indeed, let O ⊆ g be anilpotent orbit. Given ξ ∈ O and a ∈ C∗, we claim that aξ ∈ O. To this end, theJacobson-Morozov Theorem allows us to find h ∈ g for which [h, ξ] = 2ξ. Also,there exists b ∈ C for which e2b = a.

Ad(exp(bh))(ξ) = ead(bh)(ξ)

=

∞∑j=0

1

j!(ad(bh))j (ξ)

=

∞∑j=0

(2b)j

j!ξ

= e2bξ

= aξ,

where exp : g → G is the exponential map. Since Ad(exp(bh))(ξ) ∈ O, this


calculation establishes that aξ ∈ O.

Now, fix a non-zero nilpotent orbit O. The dilation action of C∗ on g commuteswith the adjoint representation ofG, and both group actions respect the inclusionO ⊆ g \ 0. In particular, P(g) = (g \ 0)/C∗ carries a residual G-action for which

P(O) := O/C∗ (2.50)

is a G-orbit.

• The Closure Order: The set N/G of nilpotent orbits carries a canonical partialorder, called the closure order. Given nilpotent orbits O1,O2 ⊆ N, one defines

O1 ≤ O2 ⇐⇒ O1 ⊆ O2. (2.51)

Strictly speaking, we should specify the topology in which the closure of O2 istaken. However, as O2 is a constructible subset of g, its Zariski and classicalclosures agree.

• The Regular Nilpotent Orbit: The poset N/G has a unique maximal elementOreg, called the regular nilpotent orbit. As the nomenclature suggests, Oreg consistsof those ξ ∈ g which are simultaneously nilpotent and regular (meaning that theLie algebra centralizer of ξ has dimension equal to the rank of g). Furthermore,Oreg is known to have a family of standard representatives. Letting eα ∈ gα \ 0,α ∈ Π, be a choice of non-zero vector in each simple root space, we have∑

α∈Π

eα ∈ Oreg (2.52)

(see [52], Lemma 5.2).

We note that Oreg is sometimes called the principal nilpotent orbit, reflecting aclose connection to Kostant’s notion of a principal three-dimensional subalge-bra (TDS). By definition, a principal TDS is a Lie subalgebra of g having the forma = spanCη, h, ξ, where ξ ∈ Oreg, η ∈ g, and h ∈ g satisfy the following condi-tions:

[ξ, η] = h, [h, ξ] = 2ξ, [h, η] = −2η.

In particular, (η, h, ξ) is an sl2(C)-triple and a is an isomorphic copy of sl2(C).

As an a ∼= sl2(C)-module, g decomposes into irreducible highest-weight repre-


sentations. Kostant showed this decomposition to be

g ∼=

r⊕j=1

V(nj) (2.53)

for some even positive integers n1, n2, . . . , nr, where r = rank(g) and V(nj) is theirreducible a-module of highest weight nj (see [52], Theorem 5.2). Perhaps sur-prisingly, Kostant found the following relationship between the representation-theoretic statement (2.53) and the Poincare polynomial of G, PG(t):

PG(t) =

r∏j=1

(1+ tnj+1).

• The Minimal Nilpotent Orbit: On the opposite extreme, one finds 0 to be theunique minimal element of N/G. While this is neither surprising nor interesting,it turns out that the set of non-zero nilpotent orbits has a unique minimal elementOmin, called the minimal nilpotent orbit.

Like the regular nilpotent orbit, Omin is known to have standard representatives.Indeed, recalling that λ denotes the highest root, one has

gλ \ 0 ⊆ Omin. (2.54)

(see [18]).

While Omin is independently interesting, it is often studied in conjunction with itsprojectivization P(Omin). This latter variety has several rich geometric structures.Indeed, P(Omin) is the G-orbit in P(g) having minimal dimension. It followsthat P(O) is a closed (hence projective) orbit, and therefore also a partial flagvariety ofG (see Proposition 9 for more details). At the same time, the symplecticstructure on Omin descends to give a complex contact structure on P(Omin). It isthen a famous conjecture of LeBrun and Salamon that minimal nilpotent orbitprojectivizations constitute the only examples of Fano contact varieties havingsecond Betti number equal to 1. This conjecture has significant implications inquaternionic Kahler geometry, where these Fano contact varieties arise as twistorspaces. For more details, we direct the reader to Chapter 6.


2.8.2 Nilpotent Orbits in Type A

We now specialize aspects of 2.8.1 to G = SLn(C) and its nilpotent orbits. Indeed, asdiscussed at the beginning of 2.8.1, X ∈ sln(C) is nilpotent in our Lie algebraic senseif and only if it is nilpotent as a matrix. Hence, by (2.1), the nilpotent orbits of SLn(C)are precisely the (SLn(C)-) conjugacy classes of nilpotent n×nmatrices. The latter areindexed by the partitions of n by means of Jordan canonical forms. In particular, onesees directly that there are only finitely many nilpotent orbits of SLn(C).

Given a partition λ = (λ1, λ2, . . . , λk) of n, let Oλ ⊆ sln(C) denote the conjugacyclass of a nilpotent Jordan matrix with k blocks of respective sizes λ1, λ2, . . . , λk. It is aclassical result of Gerstenhaber [34] that

Oλ ≤ Oµ ⇐⇒ λ ≤ µ, (2.55)

where λ and µ are compared in the dominance order (see [70]). Studying the domi-nance order, one finds that

Oreg = O(n) (2.56)

andOmin = O(2,1,1,...,1). (2.57)

Chapter 3

The T -Equivariant Cohomology ofNilpotent Orbits

In what follows, we obtain concrete, self-contained presentations of the T -equivariantcohomology rings of Oreg and Omin. Our exposition is largely based on the author’swork [20].


As discussed in 2.8.1, the study of nilpotent orbits invokes algebraic geometry, repre-sentation theory, and symplectic geometry. However, nilpotent orbits have also beenstudied in the context of algebraic topology. The work of Springer, Steinberg, andothers has led to a computation of the fundamental group of every nilpotent orbitin the classical Lie algebras (see [18]). Also, Juteau’s paper [46] gives the integralcohomology groups of the minimal nilpotent orbit in each of the finite-dimensionalcomplex simple Lie algebras. Additionally, Biswas and Chatterjee compute H2(O;R)for O any nilpotent orbit in a finite-dimensional complex simple Lie algebra (see theirpaper [10]).

We examine aspects of the equivariant algebraic topology and geometry of nilpo-tent orbits, with the following descriptions of H∗T(Oreg) and H∗T(Omin) constituting thethe main results of this section.

Theorem 7. (i) H∗T(Oreg) ∼= H∗(G/B)

(ii) Let Λ denote the collection of simple roots that are orthogonal to the highest root λ. The

28

CHAPTER 3. THE T -EQUIVARIANT COHOMOLOGY OF NILPOTENT ORBITS 29

algebra H∗T(Omin) is isomorphic to the quotient of(f[w]) ∈⊕

[w]∈W/WΛ

H∗T(pt) :(wγ)|(f[w] − f[wsγ])

∀w ∈W∀γ ∈ ∆+ s.t. 〈γ, λ〉 6= 0

by the ideal generated by (wλ)[w]∈W/WΛ .

Our arguments begin in 3.2, which provides a direct computation of H∗T(Oreg). Sec-tion 3.3 then treats the case of the minimal nilpotent orbit, but the approach differsconsiderably from that adopted when studying Oreg. In particular, we make extensiveuse of the intermediate variety P(Omin) from 2.8.1. This variety has interesting prop-erties beyond those materially relevant to computing H∗T(Omin). Notably, P(Omin) isnaturally a symplectic manifold, and the T -action on Omin descends to a Hamiltonianaction (of the real part of T ) on P(Omin). We give an explicit description of P(Omin)

T

and use it to find the moment polytope of P(Omin).

We next give a GKM-theoretic description of H∗T(P(Omin)). This algebra is thenrelated to H∗T(Omin) via a Thom-Gysin sequence in T -equivariant cohomology, com-pleting the proof of Theorem 7.

3.2 The Regular Nilpotent Orbit

Recall that an element ξ ∈ g is called regular if the dimension of the Lie algebra cen-tralizer Cg(ξ) = X ∈ g : [X, ξ] = 0 coincides with the rank of g. The regular nilpotentelements of g constitute Oreg.

Fix a point η ∈ Oreg, and let CG(η) = g ∈ G : Adg(η) = η be the G-stabilizer of η.This gives a G-equivariant isomorphism Oreg

∼= G/CG(η). Having realized Oreg in thisway, we turn our attention to CG(η). Indeed, note that Oreg is a distinguished nilpotentorbit (see [18]), meaning that the unipotent radical ofCG(η) coincides with the identitycomponent CG(η)0. It follows that

CG(η) = CG(η)0 × Z(G)

is the Levi decomposition of CG(η), where Z(G) is the (finite) centre of G. Noting thatCG(η)

0 is a copy of affine space, one has the affine bundle

EG/Z(G) → EG/CG(η)


and therefore also an isomorphism between the cohomology rings of the base and totalspaces. The ordinary cohomology of EG/CG(η) is precisely H∗G(G/CG(η)) = H

∗G(Oreg),

while that of EG/Z(G) is isomorphic to H∗Z(G)(pt). In other words,

H∗G(Oreg) ∼= H∗Z(G)(pt).

However, since Z(G) is a finite group,H∗Z(G)(pt) is isomorphic toQ (by which we meanthe graded Q-algebra equal to Q in degree zero and vanishing in all other gradingdegrees). We thus have

H∗G(Oreg) ∼= Q. (3.1)

Now, let K be a compact real form of G such that TR := K ∩ T is a real form ofT . Recall that the TR-equivariant cohomology of a smooth K-manifold is obtained bytensoring its K-equivariant cohomology with H∗TR(pt) over H∗TR(pt)W (see Proposition1(iii) of [14]). In our case, this gives

H∗T(Oreg) = H∗TR(Oreg) ∼= H∗TR(pt)⊗H∗

TR(pt)W H

∗K(Oreg). (3.2)

Using (3.1) and recalling the standard equivariant cohomology isomorphisms (2.27)and (2.30) we obtain

H∗K(Oreg) ∼= H∗G(Oreg) ∼= Q ∼= H∗K(K).

Accordingly, we may replace H∗K(Oreg) with H∗K(K) in (3.2). Applying Proposition1(iii) of [14] again, the right-hand side of (3.2) is seen to be isomorphic to H∗TR(K)

∼=

H∗(K/TR) ∼= H∗(G/B). In other words,

H∗T(Oreg) ∼= H∗(G/B),

as claimed in the statement of Theorem 7.

Remark. Without modification, our arguments establish the slightly more general factthat the T -equivariant cohomology of a distinguished nilpotent orbit coincides withthe ordinary cohomology of the full flag variety.

3.3 The Minimal Nilpotent Orbit

We now address the matter of computingH∗T(Omin). Note that we could try to proceedin analogy with Section 3.2 by fixing η ∈ Omin, taking L to be the reductive part in aLevi decomposition ofCG(η), and so forth. While this approach is certainly legitimate,


we will compute the T -equivariant cohomology of Omin by first determining that of thevariety P(Omin) mentioned in 2.8.1. With this in mind, it will be informative to givenilpotent orbit projectivizations some context in equivariant geometry.

3.3.1 The Equivariant Geometry of P(O)

Fix a non-zero nilpotent orbit O ⊆ N. Also, as before, let K be a maximal compactsubgroup of G with the property that TR := K ∩ T is a real form of T . Choose a K-invariant Hermitian inner product 〈, 〉 : g ⊗R g → C. This yields a K-invariant Kahlerstructure on P(g). Since the usual action of U(n+1) on Pn is Hamiltonian, so too is theaction of K on P(g). Furthermore, one has the moment mapΦ : P(g) → k∗ defined by

Φ([ξ])(X) = −i〈[X, ξ], ξ〉〈ξ, ξ〉

,

where X ∈ g \ 0 and η ∈ k (see [22] for a derivation of Φ). Note that the Kahlerstructure on P(g) restricts to a K-invariant Kahler structure on the smooth subvarietyP(O), and that the action of K on P(O) is Hamiltonian.

Let us take a moment to examine the Hamiltonian action of TR on P(O). We have

g = t⊕⊕β∈∆

gβ,

the weight space decomposition of the adjoint representation. Note that a point inP(g) is fixed by T if and only if it is a class of vectors in g \ 0 with the property that Tacts by scaling each vector. In other words,

P(g)T = P(t) ∪ gβ : β ∈ ∆.

With this description, we may determine P(O)T . Indeed, since t consists of semisimpleelements of g while O consists of non-zero nilpotent elements, we find that t ∩ O = ∅.Hence,

P(O)T = gβ : β ∈ ∆, gβ ∩ O 6= ∅,

a finite set. In particular, P(O)T is non-empty if and only if O is the orbit of a rootvector.

Let us take a moment to provide a more refined description of P(O)T . To this end,we will require the following lemma.


Lemma 8. Let β, γ ∈ ∆ be roots. The root spaces gβ and gγ are G-conjugate if and only if βand γ are conjugate underW.

Proof. Suppose that w ∈ W and that β = w · γ. Choosing a representative g ∈ NG(T)

of w, this means precisely that β = γ ϕg−1 |T , where ϕg−1 : G → G is conjugation byg−1. Given t ∈ T and ξ ∈ gβ, note that

Ad(t)(Ad(g)(ξ)) = Ad(g)(Ad(g−1tg)(ξ))

= Ad(g)(β(g−1tg)ξ)

= Ad(g)(γ(t)ξ)

= γ(t)(Ad(g)(ξ)).

It follows that gγ = Ad(g)(gβ).Conversely, suppose that g ∈ G and that gγ = Ad(g)(gβ). Consider the Zariski-

closed subgroupL := x ∈ G : Ad(x)(gγ) = gγ,

noting that T, gTg−1 ⊆ L. Since T and gTg−1 are maximal tori of L, there exists x ∈ L forwhich xTx−1 = gTg−1. Hence, x−1g ∈ NG(T) and Ad(x−1g)(gβ) = gγ. We may thereforeassume that g ∈ NG(T). Now, letw ∈W denote the class of g. Given t ∈ T and ξ ∈ gβ,we find that

(w · β)(t)ξ = β(g−1tg)ξ

= Ad(g−1tg)(ξ)

= Ad(g−1)(γ(t)Ad(g)(ξ))

= γ(t)ξ.

It follows that γ = w · β.

Since g is a simple Lie algebra, the root system associated with the pair (g, t) isirreducible. Hence, there are at most two distinct root lengths (namely, those of thelong and short roots), and the roots of a given length constitute an orbit ofW in ∆. ByLemma 8, there are at most two nilpotent G-orbits O for which P(O)T is non-empty,the orbits of root vectors for the short and long roots. Furthermore, if O is the orbit ofa root vector eβ ∈ gβ \ 0, β ∈ ∆, then P(O)T is the union of the points gγ for all γ ∈ ∆with length equal to that of β. Since Omin is the orbit of a highest (hence long) rootvector, P(Omin)

T = gγ : γ ∈ ∆long, where ∆long ⊆ ∆ is the set of long roots.


3.3.2 The Moment Polytope of P(Omin)

Note that the moment mapΦ : P(g) → k∗ considered earlier can be modified to obtaina moment map for the Hamiltonian action of TR on P(Omin). Indeed, we denote byµ : P(Omin) → (tR)

∗ the moment map given by the composition

P(Omin) → P(g) Φ−→ k∗ → (tR)∗

(where tR denotes the Lie algebra of TR).Recall that

P(Omin)T = gβ : β ∈ ∆long.

Given β ∈ ∆long, choose a point eβ ∈ gβ \ 0. Note that for X ∈ tR,

µ(gβ)(X) = −i〈[X, eβ], eβ〉〈eβ, eβ〉

= −ideβ(X)〈eβ, eβ〉〈eβ, eβ〉

= −ideβ(X), (3.3)

where deβ : tR → iR is the morphism of real Lie algebras induced by β : TR → U(1). Ifone regards the weight lattice X∗(T) as included into (tR)

∗ in the usual way, then (3.3)becomes

µ(gβ) = β.

The moment polytope µ(P(Omin)) is then the convex hull of ∆long in (tR)∗.

3.3.3 A Description of Omin and P(Omin)

We devote this section to giving explicit descriptions of Omin and P(Omin) as homoge-neous G-varieties. To this end, fix a highest root vector eλ ∈ gλ \ 0 ⊆ Omin, and let[eλ] ∈ P(Omin) denote its class in P(Omin). Also, let

Λ := α ∈ Π : 〈α, λ〉 = 0 (3.4)

be the collection of simple roots orthogonal to the highest root λ.

Proposition 9. TheG-stabilizerCG([eλ]) coincides with PΛ, inducing aG-equivariant varietyisomorphism G/PΛ ∼= P(Omin).

Proof. Since λ is the highest root, Ad(b)(eλ) is a non-zero multiple of eλ for each b ∈ B.It follows that B ⊆ CG([eλ]), so that CG([eλ]) is a standard parabolic subgroup of G.It will therefore suffice to prove that for α ∈ Π, g−α belongs to Cg([eλ]) if and only if〈α, λ〉 = 0, where Cg([eλ]) denotes the Lie algebra of CG([eλ]). Noting that Cg([eλ]) =


ξ ∈ g : [ξ, gλ] ⊆ gλ, and that [g−α, gλ] ∩ gλ = 0, we see that g−α ⊆ Cg([eλ]) if andonly if [g−α, gλ] = 0. So, our task is actually to prove that [g−α, gλ] = 0 if and only if〈α, λ〉 = 0.

Now, assume that [g−α, gλ] = 0. Since λ is the highest root, we also have [gα, gλ] =

0. It follows that[[g−α, gα], gλ] = 0. (3.5)

Letting hα ∈ [g−α, gα] be the coroot associated with α and regarding λ as an element oft∗, (3.5) gives

0 = [hα, eλ] = λ(hα)eλ = 2〈α, λ〉〈α,α〉

eλ.

It follows that 〈α, λ〉 = 0.Conversely, assume that 〈α, λ〉 = 0. If [g−α, gλ] 6= 0, then λ− α is a root. Hence,

sα(λ− α) = λ− 2〈α, λ〉〈α,α〉

α+ α = λ+ α

is also a root, contradicting the maximality of λ.

Let us now address the G-variety structure of Omin. To this end, we denote byL

π−→ P(g) the tautological line bundle over P(g). Recall that for ξ ∈ g \ 0, we haveπ−1([ξ]) = spanCξ. Furthermore, the tautological bundle is G-equivariant, with theG-action on the total space L given by

g : ([ξ], v) 7→ ([Adg(ξ)],Adg(v)),

g ∈ G, ξ ∈ g \ 0, v ∈ spanCξ.

Let E ϕ−→ P(Omin) denote the pullback of L along the inclusion P(Omin) → P(g).Note that Omin G-equivariantly (and alsoC∗-equivariantly) includes into E as a smoothopen subvariety, namely the complement Ex of the zero-section. Accordingly, ouranalysis of Omin will benefit from a closer examination of the bundle E. In particular,note that E inherits from L the structure of a G-equivariant line bundle over P(Omin).Having identified Omin withG/PΛ via Proposition 9, we may exhibit E as an associatedbundle for the one-dimensional PΛ-representation ϕ−1([eλ]) = gλ. Using the notation(2.17), our conclusion becomes E ∼= L(λ). With this in mind, the following is a sum-mary of our discussion.

Theorem 10. There is an isomorphism ofG-equivariant C∗-bundles overG/PΛ between Omin

and L(λ)x.


3.3.4 The T -Equivariant Cohomology of Omin

Let us use the description of Omin provided in Theorem 10 to compute H∗T(Omin). Tothis end, we have the equivariant Thom-Gysin sequence

· · ·→ Hi−2T (G/PΛ) → HiT(L(λ)) → HiT(L(λ)x) → · · · (3.6)

associated with the zero-section G/PΛ in L(λ) and its complement L(λ)x. We can sayconsiderably more about this sequence in our context, but it will require a brief com-putation of the T -equivariant first Chern class cT1(N) ∈ H2T(G/PΛ) of the normal bundleN ∼= L(λ) of the zero-section in L(λ). Recalling the notation (2.4), consider the inclu-sion i[w] : x([w]) → G/PΛ of the T -fixed point x([w]) ∈ (G/PΛ)

T , [w] ∈W/WΛ. Let

i∗[w] : H∗T(G/PΛ) → H∗T(pt)

denote the associated restriction map on T -equivariant cohomology. The followinglemma computes the restriction of cT1(L(λ)) to each T -fixed point in G/PΛ.

Lemma 11. If w ∈W, then i∗[w](cT1(L(λ)) = wλ.

Proof. Note thati∗[w](c

T1(L(λ))) = c

T1(L(λ))x([w])),

where L(λ)x([w]) is the fibre over x([w]), viewed as a T -equivariant vector bundle overa point. Recalling (2.32), the T -equivariant first Chern class of this bundle is preciselyits weight as a representation of T . To compute this weight, choose a representativek ∈ NG(T) of w. Note that by (2.12) and the discussion preceding it, any element ofL(λ)x([w]) is of the form [(k, ξ)], ξ ∈ gλ. For t ∈ T , we have

t · [(k, ξ)] = [(tk, ξ)]

= [(k(k−1tk), ξ)]

= [(k, (k−1tk) · ξ)]

= [(k, λ(k−1tk)ξ)]

= [(k, (wλ)(t)ξ)]

= (wλ)(t)[(k, ξ)].

Hence, wλ = cT1(L(λ)x([w])) = i∗[w](c

T1(L(λ))).


Now, recall thatH∗T((G/PΛ)

T) =⊕

[w]∈W/WΛ

H∗T(pt) (3.7)

as Q-algebras. Lemma 11 is then seen to imply that the image of cT1(L(λ)) under therestriction map

H∗T(G/PΛ) → H∗T((G/PΛ)T)

has a non-zero projection to each direct summand appearing in (3.7). Since restrictiongives an inclusion of H∗T(G/PΛ) into H∗T((G/PΛ)

T) as a subalgebra, and since H∗T(pt)has no zero-divisors, we conclude that cT1(L(λ)) is not a zero-divisor in H∗T(G/PΛ). Itfollows that our Thom-Gysin sequence splits into the short-exact sequences

0→ Hi−2T (G/PΛ) → HiT(L(λ)) → HiT(L(λ)x) → 0. (3.8)

(For a proof, see [4].)For a second useful refinement of our Thom-Gysin sequence, we note that restric-

tion to the zero-section gives a T -equivariant homotopy equivalence between L(λ)

and G/PΛ. It follows that the associated restriction map H∗T(L(λ)) → H∗T(G/PΛ) is anisomorphism. Using this isomorphism, we shall replace H∗T(L(λ)) in (3.8) to obtain

0→ Hi−2T (G/PΛ) → HiT(G/PΛ) → HiT(L(λ)x) → 0.

The map Hi−2T (G/PΛ) → HiT(G/PΛ) is multiplication by cT1(L(λ)) (see [9], for instance).Furthermore, the map HiT(G/PΛ) → HiT(L(λ)

x) is the map ψ∗ on equivariant cohomol-ogy induced by the projection ψ : L(λ)x → G/PΛ. (This follows from the fact that thebundle projection L(λ) → G/PΛ and zero-section G/PΛ → L(λ) give inverse maps onequivariant cohomology.)

We conclude that ψ∗ : H∗T(G/PΛ) → H∗T(Omin) is a surjective graded algebra mor-phism. Its kernel is 〈cT1(L(λ))〉, the ideal of H∗T(G/PΛ) generated by the equivariantfirst Chern class cT1(L(λ)) ∈ H2T(G/PΛ). In particular, there is a graded algebra isomor-phism

H∗T(Omin) ∼= H∗T(G/PΛ)/〈cT1(L(λ))〉.

Using Lemma 11 and Theorem 10, we obtain the second conclusion of Theorem 7.

Remark. In [46], D. Juteau used a non-equivariant version of the Thom-Gysin sequence(3.6) to help compute the ordinary integral cohomology groups of Omin. However,there is an interesting difference between the equivariant and non-equivariant cases.Indeed, Juteau found that multiplication by the ordinary first Chern class c1(L(λ)) ∈


H2(G/PΛ;Z) gave rise to a non-injective map Hi−2(G/PΛ;Z) → Hi(G/PΛ;Z) for somevalues of i. This is in contrast to the equivariant setup, as cT1(L(λ)) is not a zero-divisorin H∗T(G/PΛ).

3.3.5 An Example

Let us compute the equivariant cohomology of the minimal nilpotent orbit of G =

SL2(C). Note that ∆ = −2, 2 ⊆ Z ∼= X∗(T) is the resulting collection of roots, and that2 is the highest one. This highest root is not orthogonal to any of the simple roots, sothatΛ = ∅ and PΛ = B. The Weyl groupW is Z/2Z, and the generator acts by negationon the weight lattice. The subgroup WΛ is trivial. In particular, G/PΛ has two T -fixedpoints.

Since λ is identified with 2x ∈ Q[x] ∼= H∗T(pt), Theorem 2.46 implies that H∗T(G/PΛ)includes into H∗T(pt)⊕2 ∼= Q[x]⊕2 as the subalgebra

H∗T(G/PΛ)∼= (f1(x), f2(x)) ∈ Q[x]⊕2 : 2x|(f1(x) − f2(x))

= (f1(x), f2(x)) ∈ Q[x]⊕2 : f1(0) = f2(0).

Furthermore, Lemma 11 tells us that cT1(L(λ)) = (2x,−2x) when included into Q[x]⊕2.Hence,

H∗T(Omin) ∼=(f1(x), f2(x)) ∈ Q[x]⊕2 : f1(0) = f2(0)

〈(2x,−2x)〉.

Note that this is generated as a Q-algebra by y := [(x, 0)]. The relation is y2 = 0, sothat

H∗T(Omin) ∼= Q[y]/〈y2〉,

with y an element of grading degree two.We remark that this is consistent with our findings in Section 3.2. Indeed, if G =

SL2(C), then Omin = Oreg. Hence, H∗T(Omin) = H∗T(Oreg), and the latter is isomorphic tothe ordinary cohomology of G/B ∼= P1.

Chapter 4

Hessenberg Varieties for the MinimalNilpotent Orbit

The following chapter studies the equivariant geometry and topology of certain (Hes-senberg) subvarietiesG/B associated with Omin. This study is based on joint work withHiraku Abe, and the exposition largely adheres to our preprint [2]. While Hiraku andI were both fully participant in all aspects of this project, my principal contributionswere to the results holding in general Lie type (as opposed to those holding only inLie type A).


Hessenberg varieties form a rich and diverse family of subvarieties of the flag variety,including the flag variety itself, the Peterson variety, and Springer fibres. They areof interest to researchers in algebraic geometry [15, 23, 45, 60, 72], combinatorics [29,36, 41, 65], geometric representation theory [33, 69], and equivariant topology. Withrespect to the last of these areas, there has been a pronounced emphasis on equivariantcohomology computations for torus actions on Hessenberg varieties (see [3, 30, 40]).

This chapter studies a class of Hessenberg varieties arising from the minimal nilpo-tent orbit. To this end, recall that each highest root vector eλ ∈ gλ \ 0 belongs to theminimal nilpotent orbit of G. Accordingly, for a Hessenberg subspace H ⊆ g, we mayconsider the Hessenberg variety XH(eλ). This variety has received some attention inthe literature as an example of a highest weight Hessenberg variety (see [73]).

As is the case with nilpotent Hessenberg varieties in general, XH(eλ) is sometimessingular and reducible, and its geometry depends heavily on the choice of H. How-

38

CHAPTER 4. HESSENBERG VARIETIES FOR THE MINIMAL NILPOTENT ORBIT 39

ever, one distinguishing feature is that XH(eλ) is a union of Schubert varieties. Inparticular, it is invariant under the full maximal torus T .

While we present a wide array of results on the geometry and topology of XH(eλ),the following are our main results.

• There are explicit combinatorial procedures for determining the Poincare poly-nomial and irreducible components of XH(eλ) in Lie type A.

• The T -action renders XH(eλ) a GKM variety. Its GKM graph is the full subgraphof the GKM graph of G/Bwith vertex set w ∈W : gw−1λ ⊆ H.

• The restriction map i∗T : H∗T(G/B) → H∗T(XH(eλ)) is surjective. Its kernel isthe H∗T(pt)-submodule of H∗T(G/B) freely generated by the equivariant oppositeSchubert classes σ−

T (w) (see (2.35)) for all w ∈W with gw−1λ ∩H = 0.

We also prove a similar statement for the ordinary cohomology ring H∗(XH(eλ)).

Let us address the structure of this chapter. We begin with Section 4.2.1, whichintroduces and motivates XH(eλ) as an object of study. In 4.2.2, we use a commondescription of Hessenberg varieties in typeA to provide an explicit example of XH(eλ).

Section 4.3 seeks to introduce XH(eλ) through the lens of equivariant geometry.Specifically, 4.3.1 shows XH(eλ) to be T -invariant, and it gives a description of the T -fixed point set XH(eλ)T . Using this description, 4.3.2 computes |XH(eλ)

T |, the Eulernumber of XH(eλ). Also, 4.3.3 uses properties of XH(eλ)T to give an upper bound onthe codimension of XH(eλ) in G/B.

Section 4.4 exploits combinatorial descriptions of Hessenberg varieties in type Ato investigate the geometry of XH(eλ). Specifically, 4.4.1 computes the Poincare poly-nomial of XH(eλ) by means of the Hessenberg stair shape diagram (see Figure 4.1).Next, beginning with some partial results in type ADE, 4.4.2 and 4.4.3 introduce themodified Hessenberg stair shape to completely describe the irreducible componentsof XH(eλ) in type A.

Section 4.5 studies XH(eλ) via GKM theory. Specifically, 4.5.1 exhibits the GKMgraph of XH(eλ) as a full subgraph of the GKM graph ofG/B. In 4.5.2, we explain howone would implement this result to draw the GKM graph of XH(eλ) in typeA. We thenprovide the GKM graphs of all five such Hessenberg varieties in type A2.

Section 4.6 is devoted to the calculation ofH∗(XH(eλ)) andH∗T(XH(eλ)). Specifically,the restriction maps H∗(G/B) → H∗(XH(eλ)) and H∗T(G/B) → H∗T(XH(eλ)) are shownto be surjective with kernels generated by certain opposite Schubert classes.


4.2 Background

4.2.1 The Hessenberg Varieties of Interest

Suppose that H ⊆ g is a Hessenberg subspace, namely a b-invariant subspace of g con-taining b1. Note that

H = t⊕⊕γ∈∆H

gγ = b⊕⊕γ∈∆−

H

gγ. (4.1)

We shall call the roots in ∆H Hessenberg roots, while calling those in ∆−H negative Hessen-

berg roots.

Now, given ξ ∈ g, the subset

GH(ξ) := g ∈ G : Adg−1(ξ) ∈ H

is invariant under the right-multiplicative action of B on G. We may therefore define

XH(ξ) := GH(ξ)/B.

This is a closed (hence projective) subvariety of G/B, called a Hessenberg variety (see[23]). If ξ ∈ g is nilpotent, one calls XH(ξ) a nilpotent Hessenberg variety.

The following relationship between nilpotent orbits and nilpotent Hessenberg va-rieties will help to give context for the Hessenberg varieties studied in this chapter.

Lemma 12. If ξ and η belong to the same nilpotent G-orbit, then XH(ξ) and XH(η) are iso-morphic as varieties.

Proof. By assumption η = Adg(ξ) for some g ∈ G. Note that left-multiplication byg defines an isomorphism from GH(ξ) to GH(η). This isomorphism is B-equivariantfor the right-multiplicative action of B. Hence, the quotients GH(ξ)/B = XH(ξ) andGH(η)/B = XH(η) are isomorphic.

Fix a non-zero vector in the highest root space, eλ ∈ gλ \ 0, and consider thenilpotent Hessenberg variety XH(eλ). Noting that eλ belongs to the minimal nilpotentorbit Omin of G, Lemma 12 implies that XH(ξ) ∼= XH(eλ) for all ξ ∈ Omin. In this sense,XH(eλ) is precisely the Hessenberg variety arising from the minimal nilpotent orbit.

Letting the Hessenberg subspaceH vary, theXH(eλ) constitute an interesting familyof subvarieties of G/B. With respect to inclusion, the largest and smallest are Xg(eλ)

1We emphasize that H need not be a parabolic subalgebra of g.


and Xb(eλ), respectively. The former is easily seen to be G/B itself, while the latter isthe Springer fibre above eλ. In particular, XH(eλ) is sometimes singular and reducible.

To obtain additional examples, we will need to recall a concrete description of Hes-senberg varieties in type A.

4.2.2 Examples in Type A

Suppose that G = SLn(C), and that T are B are the subgroups of diagonal and upper-triangular matrices in SLn(C), respectively. Recall that for distinct i, j ∈ 1, 2, . . . , n,ti − tj denote the root

λ : T → C∗,

t1 0 0 . . . 0

0 t2 0 . . . 0...

...... . . . ...

0 0 0 . . . tn

7→ tit−1j . (4.2)

The highest root is then given by

λ := t1 − tn, (4.3)

and

eλ :=

0 0 . . . 0 1

0 0 . . . 0 0...

... . . . ......

0 0 . . . 0 0

is a choice of highest root vector.

Now, suppose that H ⊆ sln(C) is a Hessenberg subspace. There exists a uniqueweakly increasing function h : 1, 2, . . . , n → 1, 2, . . . , n with j ≤ h(j) for all j, suchthat

H = [aij] ∈ sln(C) : aij = 0 for i > h(j). (4.4)

Noting that (4.4) defines a bijective correspondence between the Hessenberg sub-spaces H and all such functions h, one calls these functions Hessenberg functions. Wewill represent a Hessenberg function h by listing its values, so that h = (h(1), h(2), . . . , h(n)).

Now, recall the isomorphism (2.11) between SL2(C)/B and F(1, 2, . . . , n− 1;Cn). If


h = (h(1), h(2), . . . , h(n)) is a Hessenberg function corresponding to H ⊆ sln(C), then

XH(eλ) ∼= V• ∈ F(1, 2, . . . , n− 1;Cn) : eλ(Vj) ⊆ Vh(j) for all j (4.5)

via the isomorphism (2.11).

Let us use (4.5) to describe XH(eλ) in the case where n = 3 and h = (2, 3, 3). Indeed,we have

XH(eλ) ∼= V• ∈ F(1, 2;C3) : eλ(V1) ⊆ V2.

Letting e1, e2, e3 denote the standard basis of C3, it is straightforward to see that eachV• ∈ XH(eλ) must satisfy V1 ⊆ spane1, e2 or e1 ∈ V2. Let X1 and X2 denote the subva-rieties of XH(eλ) defined by these respective conditions, so that XH(eλ) = X1∪X2. Notethat completing V1 ⊆ spane1, e2 to an element V• ∈ XH(eλ) is equivalent to specify-ing a 2-dimensional subspace V2 containing V1. Also, completing a V2 containing e1 toV• ∈ XH(eλ) amounts to specifying a 1-dimensional subspace V1 contained in V2. Fromthese observations, we see that each of X1 and X2 is isomorphic to P1 × P1. The inter-section of these subvarieties is seen to be two copies of P1 which themselves intersectin a single point.

In Section 4.5.2, we will study the above-mentioned example as a GKM variety (seeFigure 4.9).

4.3 The Equivariant Geometry of XH(eλ)

4.3.1 Algebraic Group Actions on XH(eλ)

In contrast to a general nilpotent Hessenberg variety, XH(eλ) is a union of Schubertvarieties. Equivalently, we have the following proposition.

Proposition 13. The variety XH(eλ) is invariant under the action of B on G/B.

Proof. It suffices to prove thatGH(eλ) is invariant under left-multiplication by elementsof B. To this end, suppose that b ∈ B and g ∈ GH(eλ). We have

Ad(bg)−1(eλ) = Adg−1(Adb−1(eλ)).

Since eλ belongs to the highest root space, Adb−1(eλ) is a scalar multiple of eλ. Hence,Adg−1(Adb−1(eλ)) is a scalar multiple of Adg−1(eλ), and therefore in H.


As a consequence of Proposition 13, XH(eλ) carries an action of the maximal torusT . Properties of this T -action will play an essential role in proving the main results inthis chapter. The first such property is a description of the T -fixed point set XH(eλ)T .To this end, recall that the T -fixed points of G/B are enumerated by the bijection

W∼=−→ (G/B)T (4.6)

w 7→ x(w) := [g],

where g ∈ NG(T) represents w ∈ W. Recalling the definition of the Hessenberg roots∆H (see (4.1)), we have the following description of XH(eλ)T .

Proposition 14. The T -fixed points of XH(eλ) are given by

XH(eλ)T = x(w) : w ∈W and w−1λ ∈ ∆H.

Proof. Suppose that w ∈ W is represented by g ∈ NG(T), so that x(w) = [g] ∈ G/B.Then x(w) ∈ XH(eλ) if and only if g = hb for some h ∈ GH(eλ) and b ∈ B. SinceGH(eλ) is invariant under right-multiplication by elements of B, this is equivalent tothe condition that g ∈ GH(eλ). Equivalently, Adg−1(eλ) ∈ H, which is precisely thestatement that w−1λ ∈ ∆H.

For example, suppose that G = SLn(C) and that T ⊆ SLn(C) and B ⊆ SLn(C) arethe maximal torus and Borel subgroup considered in 4.2.2, respectively. Recall thatW = Sn and let h : 1, 2, . . . , n → 1, 2 . . . , n be a Hessenberg function correspond-ing to H ⊆ sln(C). Since the highest root is as given in (4.3), Proposition 14 impliesthat x(w) ∈ XH(eλ)T if and only if the root space of w−1λ = tw−1(1) − tw−1(n) belongsto H. Noting that this root space is spanned by the matrix with entry 1 in position(w−1(1), w−1(n)) and all other entries 0, our characterization becomes

x(w) ∈ XH(eλ)T ⇐⇒ w−1(1) ≤ h(w−1(n)). (4.7)

4.3.2 The Euler Number of XH(eλ)

This section addresses the computation of |XH(eλ)T |, the Euler number of XH(eλ). Moreprecisely, we give a general formula for |XH(eλ)

T | and then specialize it to cases inwhich one can be more explicit.


To begin, Proposition 14 implies that

|XH(eλ)T | = |w ∈W : w−1λ ∈ ∆H| = |w ∈W : wλ ∈ ∆H|. (4.8)

SinceW preserves the set of long roots ∆long ⊆ ∆, (4.8) becomes

|XH(eλ)T | = |w ∈W : wλ ∈ ∆long,H|, (4.9)

where ∆long,H := α ∈ ∆long : gα ⊆ H. Noting thatW acts transitively on ∆long, we have

|w ∈W : wλ = α| = |w ∈W : wλ = λ| =|W|

|∆long|

for all α ∈ ∆long,H. Hence, (4.9) becomes

Proposition 15.

|XH(eλ)T | = |W|

|∆long,H|

|∆long|. (4.10)

We now specialize (4.10) to some particularly tractable cases. Firstly, (4.10) is seento imply that the Springer fibre Xb(eλ) contains exactly one-half of the T -fixed pointsin G/B.

Corollary 16. The Euler number of our Springer fibre Xb(eλ) is given by

|Xb(eλ)T | =

|W|

2. (4.11)

Proof. Since the number of positive long roots coincides with the number of negativelong roots, we see that |∆long,b| =

12|∆long|. The formula (4.11) then follows from (4.10).

Our second specialization of (4.10) is to the simply-laced case, in which ∆long = ∆

and ∆long,H = ∆H. Hence, |∆long| = dim(g) − rank(g) and |∆long,H| = dim(H) − rank(g),so that (4.10) reads as

Corollary 17. In the simply-laced case, we have

|XH(eλ)T | = |W|

(dim(H) − rank(g)dim(g) − rank(g)

). (4.12)

For example, if G = SLn(C), then

|XH(eλ)T | = (n− 2)!(dim(H) − n+ 1).


4.3.3 The Size of XH(eλ)

Let us write Π = α1, α2, . . . , αn, and let Πi := Π − αi for i ∈ 1, 2, . . . , n. Denoteby Pi ⊆ G be the maximal parabolic subgroup corresponding to Πi ⊆ Π and recall thatWΠi is the subgroup ofW generated by the simple reflections sαk for k 6= i.

Lemma 18. If w ∈WΠi , then w−1λ ∈ ∆+.

Proof. Note that λ =∑n

k=1mkαk with mk > 0 for k ∈ 1, 2, . . . , n. Also, recall that wehave

sα`(αk) = αk −2〈αk, α`〉〈α`, α`〉

α`. (4.13)

If w ∈ WΠi and we write w−1λ =∑n

k=1 dkαk for some dk ∈ Z, then (4.13) impliesdi = mi > 0. Since w−1λ is a root, this shows it to be a positive root.

Proposition 19. For any maximal parabolic subgroup Pi, we have Pi/B ⊆ XH(eλ) ⊆ G/B.

Proof. Lemma 18 shows that gw−1λ ⊆ b ⊆ H for any w ∈ WΠi . This means thatx(w) ∈ XH(eλ)T for all w ∈ WΠi . Since we have Pi/B =

∐w∈WΠi

Bx(w) and XH(eλ) =∐w∈XH(eλ)T Bx(w), we obtain Pi/B ⊆ XH(eλ) ⊆ G/B.

This proposition has interesting implications for the “size” of XH(eλ) in G/B. In-deed, XH(eλ)T “large” by virtue of containing a copy of WΠi for all i ∈ 1, 2, . . . , n.Secondly, whenG = SLn(C), a suitable choice of maximal parabolic Pi gives (by Propo-sition 19)

F(1, 2, . . . , n− 2;Cn−1) ⊆ XH(eλ) ⊆ F(1, 2, . . . , n− 1;Cn).

Hence, the complex codimension of XH(eλ) in F(1, 2, . . . , n − 1;Cn) is at most n − 1

when G = SLn(C). Finally, in all Lie types, it is known that the codimension of Xb(eλ)

in G/B is equal to half the dimension of the minimal nilpotent orbit Omin ( [67] cf. [17,Corollary 3.3.24]). Since dimC(Omin) = 2h

∨ − 2 (see [75])2, we have

codimC(Xb(eλ)) = h∨ − 1.

For a general Hessenberg subspace H, the inclusion Xb(eλ) ⊆ XH(eλ) gives

codimC(XH(eλ)) ≤ h∨ − 1.

2Here, h∨ is the dual Coxeter number.


4.4 Poincare Polynomials and Irreducible Components

4.4.1 Poincare Polynomials in Type A

We now compute the Poincare polynomial PH(t) of XH(eλ) whenG = SLn(C). Accord-ingly, we shall assume all notation to be as in 4.2.2.

Consider an n× n grid of boxes, and let (i, j) denote the box in row i and columnj. If H ⊆ sln(C) is a Hessenberg subspace with Hessenberg function h, we shall callthe stair-shaped sub-grid

(i, j) ∈ 1, 2, . . . , n× 1, 2, . . . , n : i ≤ h(j)

the Hessenberg stair shape. One identifies it by drawing a line in the n × n grid suchthat the sub-grid consists precisely of the boxes lying above the line.

Figure 4.1: The Hessenberg stair shape determined by h = (2, 4, 5, 5, 5) when n = 5

Definition 20. For 0 ≤ i ≤ 2n − 3, we define qH(i) to be the number of boxes inthe Hessenberg stair shape meeting the diagonal line segment joining (2, n − i) and(2+ i, n). Namely,

qH(i) = |(k, j) ∈ 1, 2, . . . , n× 1, 2, . . . , n : 2 ≤ k ≤ h(n+ 1− j), j+ k− 3 = i|.

The number qH(i) is easily computed in practice. One starts with the rightmostbox in row #2, moves i boxes to the left, and then draws the longest possible diagonalline segment passing through the current box and not passing through a box in row#1. The number of boxes meeting this line segment is precisely qH(i).

The following figure illustrates the computation of qH(2) in the case n = 5 andh = (2, 4, 5, 5, 5).


t t t@@@

Figure 4.2: The computation of qH(2) when h = (2, 4, 5, 5, 5) and n = 5. One beginsby drawing the diagonal line segment connecting (2, 3) and (4, 5). Since this segmentmeets exactly 3 boxes, we have qH(2) = 3.

It will be convenient to consider the polynomial

qH(t) :=

2n−3∑i=0

qH(i)t2i.

As is the case with its coefficients, this polynomial can be computed diagrammaticallyvia the Hessenberg stair shape. One simply fills each box involved in the computationof qH(i) with t2i, and then sums the resulting terms. The following figure illustratesthis procedure.

t0

t2

t4

t6

t2

t4

t6

t8

t4

t6

t8

t10

t6

t8

t10

t8

Figure 4.3: The computation of qH(t) for n = 5 and h = (2, 4, 5, 5, 5). One has qH(t) =1+ 2t2 + 3t4 + 4t6 + 4t8 + 2t10.

While qH(t) is not itself the Poincare polynomial of XH(eλ), we have the followingproposition.

Proposition 21. The Poincare polynomial of XH(eλ) is given by

PH(t) = qH(t)

n−3∏`=1

(1+ t2 + . . .+ t2`).

Proof. Since XH(eλ) is a union of Schubert cells, the type A fixed point criterion (4.7)


implies that

PH(t) =∑w∈Sn

w−1(1)≤h(w−1(n))

t2`(w).

Writing j = w−1(n) and k = w−1(1), we have

PH(t) =

n∑j=1

( j−1∑k=1

∑v∈Sn−2

t2`(v)+2(k−1+n−j) +

h(j)∑k=j+1

∑v∈Sn−2

t2`(v)+2(k−1+n−j−1)),

which can be explained as follows. In the one-line notation for w, if the position of 1is to the left of the position of n (so 1 ≤ k ≤ j− 1), then 1 has k− 1 inversion pairs andn has n − j inversion pairs. If the position of 1 is to the right of the position of n (soj+ 1 ≤ k ≤ h(j)), then 1 has k− 1 inversion pairs (including the pair (n, 1)) and n hasn− j− 1 inversion pairs (except for the pair (n, 1), which is already counted).

Now, note that

∑v∈Sn−2

t2`(v) =

n−3∏`=1

(1+ t2 + . . .+ t2`),

as each polynomial is the Poincare polynomial of F(1, 2, . . . , n − 3Cn−2). Hence, adirect computation gives

PH(t) =

n∑j=1

h(j)∑k=2

t2(k−2+n−j) ·n−3∏`=1

(1+ t2 + . . .+ t2`)

=

n∑j=1

h(n+1−j)∑k=2

t2(j+k−3) ·n−3∏`=1

(1+ t2 + . . .+ t2`),

and the claim follows.

Recall that for n = 5 and h = (2, 4, 5, 5, 5) we have qH(t) = 1 + 2t2 + 3t4 + 4t6 +

4t8 + 2t10. In this case, Proposition 21 yields the Poincare polynomial

PH(t) = (1+ 2t2 + 3t4 + 4t6 + 4t8 + 2t10) · (1+ t2)(1+ t2 + t4)

= 1+ 4t2 + 9t4 + 15t6 + 20t8 + 21t10 + 16t12 + 8t14 + 2t16.

Noting that h = (1, 2, . . . , n) corresponds to the Hessenberg subspace b, we obtainthe following specialization of Proposition 21.


Corollary 22. The Poincare polynomial of our Springer fiber Xb(eλ) is given by

Pb(t) = (1+ 2t2 + 3t4 + . . .+ (n− 1)t2(n−2)) ·n−3∏`=1

(1+ t2 + . . .+ t2`).

Additionally, Proposition 21 allows one to deduce the following combinatorial for-mula for dimC(XH(eλ)).

Corollary 23. The dimension of XH(eλ) is given by

dimC(XH(eλ)) =1

2(n− 1)(n− 2) + max h(j) − j | j = 1, . . . , n.

4.4.2 Irreducible Components in Type ADE

We now examine the irreducible components of XH(eλ). As one might expect, theseare precisely the maximal Schubert varieties X(w) = Bx(w) contained in XH(eλ).

Lemma 24. The irreducible components of XH(eλ) are the Schubert varieties X(w) for themaximal w ∈W satisfying w−1λ ∈ ∆H.

Proof. By Proposition 1.5 of [42], our task is to prove the following two statements.

(i) The variety XH(eλ) is a union of the X(w) for the maximal w ∈ W satisfyingw−1λ ∈ ∆H.

(ii) If w1 6= w2 are two such maximal elements, then neither X(w1) ⊆ X(w2) norX(w2) ⊆ X(w1) holds.

Since XH(eλ) is a union of Schubert varieties (see Proposition 13), one for each T -fixed point in XH(eλ), Proposition 14 allows us to write

XH(eλ) =⋃

w−1λ∈∆H

X(w) (4.14)

Furthermore, as u ≤ v if and only if X(u) ⊆ X(v), (4.14) holds if the union is takenonly over the maximal w ∈ W satisfying w−1λ ∈ ∆H. Hence, (i) is true. Of course, thefact u ≤ v⇐⇒ X(u) ⊆ X(v) also implies that (ii) is true.

Let us assume G to be of type ADE. If β ∈ ∆, then Lemma 4.4 of [73] allows one toconsider the unique maximal wβ ∈W for which w−1

β λ = β. In other words,

wβ := maxw ∈W | w−1λ = β. (4.15)


Note that if w−1λ ∈ ∆H, then w ≤ wβ for some β ∈ ∆H (e.g. take β = w−1λ). It followsthat the maximal elements of w ∈ W | w−1λ ∈ ∆H (the set discussed in Lemma 24)are precisely the maximal elements of

ΩH := wβ : β ∈ ∆H. (4.16)

Using Lemma 24, it follows that the maximal elements of ΩH label the irreduciblecomponents of XH(eλ). However, ΩH still contains non-maximal elements, and thedetermination of its maximal elements will involve a few properties of the wβ. Tothis end, the following proposition is an immediate consequence of the discussion inSection 4 of [73].

Proposition 25. Suppose that β, γ ∈ ∆.

(i) β = γ⇐⇒ wβ = wγ

(ii) If β and γ have the same sign, then β ≤ γ⇐⇒ wγ ≤ wβ.

Corollary 26. (i) If α,β ∈ Π are distinct simple roots, then wα and wβ are incomparablein the Bruhat order.

(ii) If γ and δ are distinct minimal elements of ∆−H, then wγ and wδ are incomparable in the

Bruhat order.

(iii) Suppose that γ ∈ ∆H. If wγ is a maximal element of ΩH, then γ ∈ Π or γ is a minimalelement of ∆−

H.

Proof. Recognizing (i) and (ii) as immediate consequences of Proposition 25, we proveonly (iii). To this end, if γ is positive, then there exists α ∈ Π such that α ≤ γ. Propo-sition 25 implies that wγ ≤ wα, and the maximality of wγ then yields wγ = wα. Itfollows that γ = α is simple.

Now, assume that γ is negative and let δ ∈ ∆−H satisfy δ ≤ γ. Proposition 25 implies

wγ ≤ wδ. Since wγ is maximal, wγ = wδ and we conclude that γ = δ. It follows that γis a minimal element of ∆−

H.

In light of Corollary 26, the maximal elements ofΩH are of the following two types:

1. wα, where α ∈ Π and wα ≮ wγ for all γ ∈ ∆−H, 3

3Strictly speaking, Corollary 26 gives the following different-looking description of the maximalwα’s: wα, where α ∈ Π and wα ≮ wγ for all minimal γ ∈ ∆−

H. However, by appealing to Proposition25, one sees that this is equivalent to the description we have given.


2. wγ, where γ is a minimal element of ∆−H and wγ ≮ wα for all α ∈ Π.

In order to refine (2), we will need the following two results.

Lemma 27. If α ∈ Π, then wα < w−α.

Proof. Since (wαsα)−1λ = −α, the maximality of w−α implies that

wαsα ≤ w−α. (4.17)

Furthermore, as (wαsα)α = −λ ∈ ∆−, we have

wα < wαsα. (4.18)

By combining (4.17) and (4.18), we obtain the desired result.

Corollary 28. If α ∈ Π and γ ∈ ∆−, then wγ ≮ wα.

Proof. If wγ < wα, then Lemma 27 implies that wγ < w−α. Proposition 25 then yields−α < γ, which is impossible.

In light of the above, we have the following improved description of the maximalelements ofΩH:

1. wα, where α ∈ Π and wα ≮ wγ for all γ ∈ ∆−H

2. wγ, where γ is a minimal element of ∆−H.

Remembering that the maximal elements of ΩH label the irreducible componentsof XH(eλ), we have the following immediate corollary.

Corollary 29. If γ is a minimal element of ∆−H, then X(wγ) is an irreducible component of

XH(eλ).

The next section gives a combinatorial enumeration of the maximal elements ofΩH (and therefore also the irreducible components of XH(eλ)) in Lie type An−1.


4.4.3 Complete Description of the Irreducible Components in Type

A

Let β = ti − tj ∈ ∆. Recall from (4.3), (4.7), and (4.15) that wβ ∈ Sn is the longestpermutation satisfying wβ(i) = 1 and wβ(j) = n. For a simple root α = tj−1 − tj, wehave wα(j− 1) = 1 and wα(j) = n, i.e. the one-line notation for wα is

wα = · · · 1 n · · ·

where 1 is in the (j− 1)-st position, n is in the j-th position, and the rest of the orderedsequence wα(1), . . . , wα(j − 2), wα(j + 1), . . . , wα(n) is given by n − 1, n − 2, . . . , 3, 2.For γ = tk − t` (k > `) a negative root, the one-line notation for wγ is

wγ = · · · n · · · 1 · · · ,

where n is in the `-th position and 1 is in the k-th position.Continuing with our specialization to type An−1, we will need to introduce the

modified Hessenberg function and the modified Hessenberg stair shape. To this end, let h bea Hessenberg function. We define a function h : 1, 2, . . . , n → 1, 2, . . . , n by

h(j) :=

h(j) − 1(= j− 1) if h(j− 1) = j− 1 and h(j) = j,

h(j) otherwise,(4.19)

and we call h a modified Hessenberg function. Note that while h is a weakly increasingfunction, it might not be an honest Hessenberg function.

As with a Hessenberg function, one can consider the stair-shaped sub-grid

(i, j) ∈ 1, 2, . . . , n× 1, 2, . . . , n : i ≤ h(j),

called the modified Hessenberg stair shape (see Figure 4.4).


Figure 4.4: A Hessenberg stair shape and its modified Hessenberg stair shape

Lemma 30. If tk − t` ∈ ∆−, then

tk − t` ∈ ∆−H if and only if k ≤ h(`).

Proof. The condition tk − t` ∈ ∆−H is equivalent to gα ⊂ H, where α = tk − t`. Also,

the latter condition is equivalent to k ≤ h(`) via the correspondence (4.4) betweenHessenberg functions and Hessenberg subspaces. Thus it suffices to show that fork > `, the condition k ≤ h(`) is equivalent to k ≤ h(`).

If k ≤ h(`), then the assumption k > ` implies ` < k ≤ h(`), and hence h(`) ≥ `+ 1.This implies h(`) = h(`), and hence we obtain k ≤ h(`). If k ≤ h(`), then the definitionof h gives h(`) ≤ h(`) and we obtain k ≤ h(`).

Definition 31. For i, j ∈ 1, 2, . . . , n, we shall call (i, j) a corner of the modified Hes-senberg stair shape if i = h(j) and h(j− 1) < h(j), with the convention h(0) := 0.

Lemma 32. If (i, j) is a corner of the modified Hessenberg stair shape, then h(j) 6= j.

Proof. If h(j) = j, then in particular h(j) 6= j − 1. The definition (4.19) then impliesh(j) = h(j) = j. Now, (4.19) and h(j) 6= j − 1 also imply that the two conditionsh(j− 1) = j− 1 and h(j) = j cannot hold simultaneously. We therefore have h(j− 1) 6=j − 1, which together with h(j − 1) ≤ h(j) = j implies h(j − 1) = j. So we haveh(j− 1) = j and h(j) = j, and it follows that (i, j) cannot be a corner.

Lemma 33. For a simple root α = tj−1 − tj (2 ≤ j ≤ n), (j− 1, j) is a corner of the modifiedHessenberg stair shape if and only if wα 6< wγ for all γ ∈ ∆−

H.

Proof. To begin, assume that (j − 1, j) is a corner. By definition, we have h(j) = j − 1.Suppose in addition that there exists γ = tk−t` ∈ ∆−

H satisfyingwα < wγ. In principle,we have the following three cases.


(i) j < ` :

wα = · · · 1 n · · · · · · · · · · · · ·,

wγ = · · · · · · · · · n · · · 1 · · · .

(ii) ` ≤ j and j− 1 ≤ k :

wα = · · · · · · · · · 1 n · · · · · · ·,

wγ = · · · · · · n · · · · · · 1 · · · .

(iii) k < j− 1 :

wα = · · · · · · · · · · · · 1 n · · · ·,

wγ = · · · n · · · 1 · · · · · · · · · .

Case (i) cannot occur, since transposing n to its right is a length-decreasing process.Similarly, Case (iii) cannot occur, since transposing 1 to its left is length-decreasing.Hence, we must have ` ≤ j and j − 1 ≤ k. However, as tk − t` is not positive, one ofthe first two inequalities is strict. Hence, by Lemma 30

j− 1 ≤ k ≤ h(`) ≤ h(j) = j− 1.

However, since (h(j), j) is a corner, one of these inequalities must be strict. This is acontradiction, completing the first half of our proof.

We now prove the converse. Suppose that there is no γ ∈ ∆−H satisfying wα < wγ.

We claim that

h(j− 2) = j− 2, h(j− 1) = j− 1, and h(j) = j,

with the convention h(0) := 0. The first of these can be proved as follows. Since thecase of j = 2 is clear, we can assume j ≥ 3. If h(j− 2) ≥ j− 1, then γ := th(j−2) − tj−2 =

th(j−2) − tj−2 is a negative Hessenberg root by Lemma 30, and wα < wγ since we are inCase (ii). So h(j − 2) = j − 2 follows. The same argument proves h(j − 1) = j − 1 andh(j) = j. Now from the definition of h, we obtain

h(j− 1) = j− 2 and h(j) = j− 1.

Hence (j− 1, j) is a corner of the modified Hessenberg stair shape.


Now, recall the definition of ΩH from (4.16), as well as the description of the max-imal elements of ΩH given at the end of 4.4.2. With these considerations in mind,Lemma 33 may be restated in the following way: If α = tj−1 − tj is a simple root, thenwα is a maximal element ofΩH if and only if (j− 1, j) is a corner of the modified Hes-senberg stair shape. This is consistent with the following more complete descriptionof the maximal elements of ΩH in type A. We remind the reader that a corner of themodified Hessenberg stair shape cannot lie on the diagonal (see Lemma 32).

Proposition 34. If i, j ∈ 1, 2, . . . , n with i 6= j and β = ti − tj, then wβ is a maximalelement ofΩH if and only if (i, j) is a corner of the modified Hessenberg stair shape.

Proof. To prove the backward implication, assume that (i, j) is a corner. We shall dis-tinguish between the cases h(j) = j − 1 and h(j) 6= j − 1. In the former, (i, j) being acorner implies that i = h(j) = j− 1 (so j− 1 ≥ 1). In particular, β = ti− tj = tj−1− tj isa simple root. Lemma 33 then implies that wβ ≮ wγ for all γ ∈ ∆−

H. By the discussionat the end of 4.4.2, wβ is a maximal element ofΩH.

For our second case, suppose that h(j) 6= j − 1. Since (i, j) is a corner, Lemma 32implies that h(j) > j. Again, since (i, j) is a corner, i = h(j). In particular, i > j andti − tj is a negative Hessenberg root. As (i, j) is a corner with i > j, an application ofLemma 30 establishes that ti − tj is a minimal element of ∆−

H. The discussion at theend of 4.4.2 then shows that wβ is a maximal element ofΩH.

We now prove the forward implication. Firstly, assume that β = ti − tj is simple(so i = j − 1). By Lemma 33, (i, j) = (j − 1, j) is a corner of the modified Hessenbergstair shape.

Secondly, assume that β = ti − tj is a minimal element of ∆−H (so i > j ≥ 1). We

have h(j) = i, since th(j) − tj would otherwise be a strictly less than ti − tj. A similarargument establishes that h(j− 1) ≤ i− 1must also hold, so that (i, j) is a corner.

As noted earlier, the irreducible components of XH(eλ) correspond to the maximalelements of ΩH. Noting that these maximal elements are described in Proposition 34,the following theorem gives the irreducible components of XH(eλ) in Lie type An−1

Theorem 35. In type An−1, there is a bijective correspondence between the set of corners ofthe modified Hessenberg stair shape and the set of irreducible components of XH(eλ) given by

(h(j), j) 7→ X(wj) = Bx(wj),

where wj is the longest permutation satisfying wj(h(j)) = 1 and wj(j) = n.


Let us implement Theorem 35 in the context of a specific example. Indeed, re-call that Figure 4.4 includes the modified Hessenberg stair shape determined by h =

(2, 2, 3, 5, 6, 6, 7, 8) when n = 8. The corners are (2, 1), (5, 4), (6, 5), (7, 8), as is indicatedin the following diagram.

tt t t

Figure 4.5: The modified Hessenberg stair shape for h = (2, 2, 3, 5, 6, 6, 7, 8) with dotslabeling corners

By Theorem 35, the irreducible components of XH(eλ) are the Schubert varietiesX(w) for the following elements w ∈ S8:

8 1 7 6 5 4 3 2, 7 6 5 8 1 4 3 2, 7 6 5 4 8 1 3 2, 7 6 5 4 3 2 1 8

.

4.5 GKM Theory on XH(eλ)

We devote this section to the construction and examination of a GKM variety structureon XH(eλ).

4.5.1 The GKM Graph of XH(eλ)

Since Proposition 13 shows XH(eλ) to be a union of Schubert cells, this variety hastrivial cohomology in odd grading degrees. It follows that XH(eλ) is T -equivariantlyformal (see [35]), and hence that the GKM structure onG/B restricts to such a structureon XH(eλ). Accordingly, we will describe H∗T(XH(eλ)) by exhibiting the GKM graphof XH(eλ) as a subgraph of the GKM graph of G/B. Noting that the vertices of oursubgraph have been determined by Proposition 14, we need only determine the edges.For this latter part, we will need to briefly discuss root strings.


If α,β ∈ ∆ are roots, one has the root string

S(β,α) := (∆ ∪ 0) ∩ β+ nα : n ∈ Z.

If p and q are maximal for the properties β + pα ∈ S(β,α) and β − qα ∈ S(β,α),respectively, then

S(β,α) = β+ nα : −q ≤ n ≤ p

andq− p =

2(β,α)

(α,α)(4.20)

(see Proposition 2.29 of [49]). The relevance of root strings to our present work iscaptured by the following lemma.

Lemma 36. If w ∈W and α ∈ ∆+ are such that x(wsα) ∈ XH(eλ)T , then⊕β∈S(w−1λ,α)

gβ ⊆ H.4

Proof. First note that eitherα ∈ w−1∆+ or −α ∈ w−1∆+. Since S(w−1λ, α) = S(w−1λ,−α),we may assume that α ∈ w−1∆+ (ie. that wα ∈ ∆+). Noting that λ is the highest root,(4.20) implies that

S(λ,wα) = λ− n(wα), λ− (n− 1)(wα), . . . , λ−wα, λ

for n = 2(λ,wα)(wα,wα)

= 2(w−1λ,α)(α,α)

. Hence, S(w−1λ, α) is given by

S(w−1λ, α) = w−1λ− nα,w−1λ− (n− 1)α, . . . , w−1λ− α,w−1λ.

The lowest root in this string is w−1λ− nα = sα(w−1λ). Also, applying Proposition 14

to the condition x(wsα) ∈ XH(eλ)T gives

gw−1λ−nα = gsα(w−1λ) ⊆ H.

Since H is b-invariant, repeated bracketing with gα ⊆ b establishes that the root spaceof each root in S(w−1λ, α) lies in H. This completes the proof.

Theorem 37. The GKM graph of XH(eλ) is a full subgraph of the GKM graph of G/B.

Proof. Equivalently, we claim that if w ∈ W and α ∈ ∆+ are such that xw, x(wsα) ∈4Here, it is understood that g0 = [g−α, gα].


XH(eλ)T , then Yw,α ⊆ XH(eλ) (where Yw,α is the T -invariant copy of P1 defined in (2.40)).

To this end, fix a representative g ∈ NG(T) of w, and let N−α denote the connectedclosed subgroup of SL2(C)α with Lie algebra g−α. Note that

Ye,α = N−αxe,

the closure of the N−α-orbit through xe. Furthermore, as Yw,α = gYe,α (see (2.40)), wehave

Yw,α = gYe,α = (gN−α)xe.

Since XH(eλ) is a closed subvariety of G/B, proving that (gN−α)xe ⊆ XH(eλ) will estab-lish that Yw,α ⊆ XH(eλ). To prove the former, it will suffice to establish that gh ∈ GH(eλ)for all h ∈ N−α, namely Ad(gh)−1(eλ) ∈ H.

Suppose that h ∈ N−α and note that

Ad(gh)−1(eλ) = Adh−1(Adg−1(eλ)).

Writing Adg−1(eλ) = ew−1λ ∈ gw−1λ and h = exp(ξ) for ξ ∈ g−α, we obtain

Ad(gh)−1(eλ) = Adexp(−ξ)(ew−1λ) = ead−ξ(ew−1λ) =

∞∑k=0

1

k!(ad−ξ)

k(ew−1λ) (4.21)

Furthermore, if (ad−ξ)k(ew−1λ) 6= 0, then it belongs to a root space for a root in S(w−1λ, α).

Hence,

Ad(gh)−1(eλ) =

∞∑k=0

1

k!(ad−ξ)

k(ew−1λ) ∈⊕

β∈S(w−1λ,α)

gβ.

By Lemma 36, it follows that Ad(gh)−1(eλ) ∈ H.

4.5.2 GKM Graphs of XH(eλ) in Type A

By Theorem 37, finding the GKM graph of XH(eλ) amounts to determining its T -fixedpoints. With this in mind, suppose that G = SL3(C) and that T ⊆ SL3(C) and B ⊆SL3(C) are the maximal torus and Borel considered in 4.2.2, respectively. Recall thatW = S3 and let h : 1, 2, 3 → 1, 2, 3 be a Hessenberg function corresponding toH ⊆ sl3(C).

The possible Hessenberg functions are (1, 2, 3), (1, 3, 3), (2, 2, 3), (2, 3, 3), and (3, 3, 3).By applying (4.7), one determines the T -fixed points for each corresponding varietyXH(eλ). Noting that Theorem 37 then determines the GKM graph of each variety as a


subgraph of Figure 2.1, the following are the GKM graphs of all the XH(eλ) in type A2.

tt tHH

HHH

1 2 3

1 3 22 1 3 γ12 γ23

Figure 4.6: The GKM graph of XH(eλ) for h = (1, 2, 3)

t

tt taaaaaaaaaaHHHH

H

1 2 3

2 3 1

1 3 22 1 3 γ12 γ23

γ13γ12


t

tt t

!!!!!!!!!!HHH

HH

1 2 3

1 3 2

3 1 2

2 1 3 γ12 γ23

γ13γ23


t

t tt taaaaaaaaaa

!!!!!!!!!!HHHHH

1 2 3

2 3 1

1 3 2

3 1 2

2 1 3 γ12 γ23

γ13 γ13

γ12 γ23



t

t

t tt t

HHHH

Haaaaaaaaaa

!!!!!!!!!!HHHH

H

1 2 3

3 2 1

2 3 1

1 3 2

3 1 2

2 1 3 γ12 γ23

γ13 γ13

γ23 γ12

γ23 γ12

γ13


4.6 Cohomology Ring Presentations

In this section, we use the restriction maps

i∗ : H∗(G/B) → H∗(XH(eλ)) and i∗T : H∗T(G/B) → H∗T(XH(eλ))

to explicitly presentH∗(XH(eλ)) andH∗T(XH(eλ)) as quotients ofH∗(G/B) andH∗T(G/B),respectively.

4.6.1 Ordinary Cohomology

We begin with the following proposition.

Proposition 38. The restriction map i∗ : H∗(G/B) → H∗(XH(eλ)) is surjective.

Proof. Since rational singular cohomology is the dual of rational singular homology, itsuffices to show that the map H∗(XH(eλ)) → H∗(G/B) is injective. Now, consider thecommutative diagram

H∗(XH(eλ))∼=−−−→ H∗(XH(eλ))y y

H∗(G/B)∼=−−−→ H∗(G/B)

, (4.22)

where H∗ denotes Borel-Moore homology (see [31]). The vertical maps in (4.22) arethe maps induced by the inclusion XH(eλ) → G/B, and the horizontal isomorphismsare the ones described in 6.10.14 of [68]. So what we need check is that the inducedmap H∗(XH(eλ)) → H∗(G/B) is injective. To this end, consider the subsets

(G/B)p :=∐

w∈W, `(w)≤p

Bx(w)


and(XH(eλ))p :=

∐w∈W, `(w)≤px(w)∈XH(eλ)T

Bx(w).

Since XH(eλ) is a union of Schubert cells, we have the affine pavings

G/B = (G/B)dimC(G/B) ⊇ . . . ⊇ (G/B)1 ⊇ (G/B)0 = ∅

andXH(eλ) = (XH(eλ))dimC(XH(eλ)) ⊇ . . . ⊇ (XH(eλ))1 ⊇ (XH(eλ))0 = ∅.

Also, for each p = 0, 1, . . . ,dimC(G/B), we have a commutative diagram (c.f. [31])

0 −−−→ H∗((XH(eλ))p−1) −−−→ H∗((XH(eλ))p) −−−→ ⊕`(w)=p

x(w)∈XH(eλ)T

H∗(Bx(w)) −−−→ 0

y y y0 −−−→ H∗((G/B)p−1) −−−→ H∗((G/B)p) −−−→ ⊕

`(w)=p

H∗(Bx(w)) −−−→ 0.

The left and the middle vertical maps are those induced by the inclusions, and eachcomponent of the right vertical map is the compositionH∗(Bx(w)) → H∗(

∐`(w)=p Bx(w)) →

H∗(Bx(w)). It is straightforward to see that each component map is an isomorphismHence, the right vertical map is an injection, and we see that the induced mapH∗(XH(eλ)) →H∗(G/B) is injective by induction on p. This completes the proof.

By Proposition 38, the map i∗ : H∗(G/B) → H∗(XH(eλ)) is surjective. We shalltherefore address ourselves to computing its kernel. To this end, we will need thefollowing proposition.

Proposition 39. Suppose that v,w ∈ W satisfy v ≥ w. If x(w) 6∈ XH(eλ)T , then x(v) 6∈XH(eλ)

T .

Proof. Suppose that x(v) ∈ XH(eλ)T . Since XH(eλ) is a closed B-invariant subvarietyof G/B, it follows that X(v) = Bx(v) ⊆ XH(eλ). Noting that v ≥ w, we must havex(w) ∈ X(v). This contradicts our assumption that x(w) 6∈ XH(eλ)T .

Now, recall the definition of the opposite Schubert variety X−(w) := Bx(w), w ∈W , as well as the definition of the corresponding opposite Schubert class σ−(w) ∈H2`(w)(G/B).


Corollary 40. The subsetJH :=

⊕x(w)6∈XH(eλ)T

Qσ−(w) (4.23)

is an ideal of H∗(G/B).

Proof. If u ∈W and x(w) /∈ XH(e)T , then ordinary Schubert calculus gives

σ−(u)σ−(w) =∑v≥u,w

cvuwσ−(v) (4.24)

for some cvuw ∈ Z. By Proposition 39, x(v) 6∈ XH(eλ)T for all v appearing in the sum(4.24). Hence, σ−(u)σ−(w) ∈ JH, proving that JH is an ideal.

With these considerations in mind, we offer the following presentation ofH∗(XH(eλ)).

Theorem 41. The map i∗ induces a graded Q-algebra isomorphism

H∗(G/B)/JH → H∗(XH(eλ)).

Proof. To begin, we claim that i∗(σ−(w)) = 0 for w ∈ W satisfying x(w) 6∈ XH(eλ)T .This will follow from our establishing that

X−(w) ∩ XH(eλ) = ∅. (4.25)

To this end, we have

X−(w) =∐w≤v

B−x(v) and XH(eλ) =∐

x(u)∈XH(eλ)TBx(u), (4.26)

with the latter decomposition being a consequence of Proposition 13. Now, recall thatfor u, v ∈ W, B−x(v) ∩ Bx(u) 6= ∅ if and only if v ≤ u. So, if X−(w) ∩ XH(eλ) 6= ∅,then 4.26 implies w ≤ u for some u ∈ W with x(u) ∈ XH(eλ)T . Proposition 39 thengives x(w) ∈ XH(eλ)T , which is a contradiction. We conclude that 4.25 holds, so thati∗(σ−(w)) = 0whenever x(w) 6∈ XH(eλ)T .

In light of our findings, i∗ induces a surjective graded Q-algebra homomorphism

H∗(G/B)/JH → H∗(XH(eλ)). (4.27)

To conclude that (4.27) is an isomorphism, it will suffice to prove that

dimQ(H∗(G/B)/JH) = dimQ(H

∗(XH(eλ))). (4.28)


Noting that

H∗(G/B) =⊕w∈W

Qσ−(w),

we have dimQ(H∗(G/B)/JH) = |XH(eλ)

T |. Also, the Schubert cell decomposition ofXH(eλ) gives dimQ(H

∗(XH(eλ))) = |XH(eλ)T |. Hence, (4.28) is satisfied and the map

H∗(G/B)/JH → H∗(XH(eλ)) is an isomorphism.

For example, suppose that G = SL3(C) and that all notation is as presented in4.2.2. We will use Theorem 41 to obtain a presentation of H∗(Xb(eλ)). To this end, letF(1, 2;C3)× C3 be the trivial vector bundle over F(1, 2;C3), and set

Ei := (V•, v) ∈ F(1, 2;C3)× C3 | v ∈ Vi

for i ∈ 1, 2, 3. Note that Ei is a complex vector bundle over F(1, 2;C3). Each quotientLi := Ei/Ei−1 is a complex line bundle, allowing us to consider its first Chern class

c1(Li) ∈ H∗(F(1, 2;C3)).

Now, recall that the algebra morphism

Q[x1, x2, x3] → H∗(F(1, 2;C3)), xi 7→ c1(Li), i ∈ 1, 2, 3

is surjective. Recall also that its kernel is the ideal generated by e1(x), e2(x), ande3(x), where ei(x) denotes the i-th elementary symmetric polynomial in the variablesx1, x2, x3. In particular, we have an algebra isomorphism

H∗(F(1, 2;C3))∼=−→ Q[x1, x2, x3]/(e1(x), e2(x), e3(x)). (4.29)

The ideal Jb ⊆ H∗(F(1, 2;C3)) is seen to be generated by the opposite Schubert classesσ−(2 3 1), σ−(3 1 2), σ−(3 2 1) ∈ H∗(F(1, 2;C3)). Their images under the isomorphism(4.29) are

σ−(2 3 1) = x1x2, σ−(3 1 2) = x1x1, σ−(3 2 1) = x1x1x2,

where (by an abuse of notation) xi is also used to denote its image in the quotient


algebra Q[x1, x2, x3]/(e1(x), e2(x), e3(x)). Applying Theorem 41, we obtain

H∗(Xb(eλ)) ∼= H∗(G/B)/Jb ∼=Q[x1, x2, x3]/(e1(x), e2(x), e3(x))Qx1x2 ⊕Qx1x1 ⊕Qx1x1x2

.

A straightforward manipulation of the rightmost ring then yields

H∗(Xb(eλ)) ∼= Q[x1, x2, x3]/(e1(x), e2(x), e3(x), x1x2, x1x3, x2x3),

which is exactly Tanisaki’s presentation of H∗(Xb(eλ)) (see [71]).

4.6.2 Equivariant Cohomology

As one might expect, we have the following equivariant counterpart of Proposition38.

Proposition 42. The restriction map i∗T : H∗T(G/B) → H∗T(XH(eλ)) is surjective.

Proof. Consider the commutative diagram

0 −−−→ H>0T (pt) −−−→ H∗T(G/B) −−−→ H∗(G/B) −−−→ 0

id

y i∗T

y i∗

y0 −−−→ H>0T (pt) −−−→ H∗T(XH(eλ)) −−−→ H∗(XH(eλ)) −−−→ 0,

where H>0T (pt) is the ideal of H∗T(pt) generated by the elements of positive degree (see2.5.2). The surjectivity of i∗T then follows from that of i∗.

Proceeding in analogy with 4.6.1, we now compute the kernel of i∗T . To this end,recall definition (2.35) of the equivariant opposite Schubert class σ−

T (w). We will needthe following well-known description of the image of σ−

T (w) under the restriction map

i∗w : H∗T(G/B) → H∗T(x(w)) = H∗T(pt). (4.30)

Lemma 43. If w ∈W, then

i∗w(σ−T (w)) =

∏α∈∆+∩w∆−

α. (4.31)

Proof. Since x(w) is a smooth point of X−(w), i∗w(σ−T (w)) is precisely the T -equivariant

Euler class of the T -representation

Tx(w)(G/B)/Tx(w)(X−(w)). (4.32)


It is therefore equal to the product of the weights occurring in (4.32), which we nowdetermine. To this end, as wBw−1 is the G-stabilizer of x(w) and has Lie algebra wb,we have isomorphisms

Tx(w)(G/B) ∼= g/wb ∼=⊕α∈w∆−

gα (4.33)

of T -representations. Also, the B−-stabilizer of x(w) is B−∩wBw−1 and has Lie algebrab− ∩wb. We therefore have

Tx(w)(X−(w)) = Tx(w)(B−wB/B) ∼= b−/(b− ∩wb) ∼=⊕

α∈∆−∩w∆−

gα. (4.34)

Combining (4.33) and (4.34), one finds that

Tx(w)(G/B)/Tx(w)(X−(w)) ∼=⊕

α∈∆+∩w∆−

gα

as T -representations. This completes the proof.

These equivariant opposite Schubert classes are seen to form an H∗T(pt)-modulebasis of H∗T(G/B). With this in mind, the following corollary introduces an importantH∗T(pt)-submodule of H∗T(G/B).

Corollary 44. The submodule

JTH :=⊕

x(w) 6∈XH(eλ)TH∗T(pt)σ−

T (w) (4.35)

is an ideal of H∗T(G/B).

Proof. The argument is similar to that used in the proof of Proposition 40, providedone uses the well-known fact that

σ−T (u)σ

−T (w) =

∑v≥u,w

cvuwσ−T (v) (4.36)

for cvuw ∈ H∗T(pt). For the reader’s convenience, we briefly recount the proof of thisfact. To this end, let v ∈W be a minimal element with the property that

i∗v(σ−T (u)σ

−T (w)) 6= 0. (4.37)


Note that vsα < v for all α ∈ ∆+ ∩ v−1∆−, so that

i∗vsα(σ−T (u)σ

−T (w)) = 0, α ∈ ∆+ ∩ v−1∆−.

The GKM conditions defining the image of H∗T(G/B) → H∗T((G/B)T) (see (2.46)) then

give(vα) | i∗v(σ

−T (u)σ

−T (w)), α ∈ ∆+ ∩ v−1∆−.

Hence, the product ∏α∈∆+∩v−1∆−

vα (4.38)

also divides i∗v(σ−T (u)σ

−T (w)). Using Lemma 43, one finds that (4.38) coincides with

(−1)l(v)i∗v(σ−T (v)). In particular, i∗v(σ

−T (v)) divides i∗v(σ

−T (u)σ

−T (w)), meaning that

i∗v(σ−T (u)σ

−T (w) − c

vuwσ

−T (v)) = 0 (4.39)

for some cvuw ∈ H∗T(pt).Continuing the support-reducing process by induction, one eventually obtains a

class with no support in the GKM graph. In other words, there exist coefficients cvuw ∈H∗T(pt) for all v ≥ u,w such that

σ−T (u)σ

−T (w) −

∑v≥u,w

cvuwσ−T (v)

has zero image under the localization map H∗T(G/B) → H∗T((G/B)T). Since the local-

ization map is injective, we conclude that

σ−T (u)σ

−T (w) =

∑v≥u,w

cvuwσ−T (v). (4.40)

Theorem 45. The map i∗T : H∗T(G/B) → H∗T(XH(eλ)) induces a graded H∗T(pt)-algebra iso-morphism

H∗T(G/B)/JTH → H∗T(XH(eλ)).

Proof. In the proof of Theorem 41, we showed that i∗T(σ−T (w)) = 0 for x(w) 6∈ XH(eλ)T .

Therefore, i∗T induces a surjective map

H∗T(G/B)/JTH → H∗T(XH(eλ)). (4.41)


Now, from the definition of JTH, it is clear that

H∗T(G/B)/JTH∼=

⊕x(w)∈XH(eλ)T

H∗T(pt)σ−T (w)

as H∗T(pt)-modules. In particular, H∗T(G/B)/JTH is free of rank |XH(eλ)

T |. However, asXH(eλ) is T -equivariantly formal, H∗T(XH(eλ)) is also free of rank |XH(eλ)

T |. It followsthat (4.41) is actually an isomorphism.

Chapter 5

Nilpotent Orbit Complexification

The following chapter introduces and studies nilpotent orbit complexification, a mech-anism for relating real and complex nilpotent orbits. The content is largely based onthe preprint [19].

Please note that we will deviate significantly from the conventions and notation inChapter 2.


Real nilpotent orbits have been studied in a variety of contexts, including differentialgeometry, symplectic geometry, and Hodge theory (see [63]). Attention has also beengiven to the interplay between real and complex nilpotent orbits, with the Kostant-Sekiguchi Correspondence (see [64]) being perhaps the most famous instance. Accord-ingly, we provide additional points of comparison between real and complex nilpotentorbits. Specifically, let g be a finite-dimensional semisimple real Lie algebra with com-plexification gC. Each real nilpotent orbit O ⊆ g lies in a unique complex nilpotentorbit OC ⊆ gC, the complexification of O. The following is our main result.

Theorem 46. The process of nilpotent orbit complexification has the following properties.

(i) Every complex nilpotent orbit is realizable as the complexification of a real nilpotent orbitif and only if g is quasi-split and has no simple summand of the form so(2n+1, 2n−1).

(ii) If g is quasi-split, then a complex nilpotent orbit Θ ⊆ gC is realizable as the complex-ification of a real nilpotent orbit if and only if Θ is invariant under conjugation withrespect to the real form g ⊆ gC.

68

CHAPTER 5. NILPOTENT ORBIT COMPLEXIFICATION 69

(iii) If O1,O2 ⊆ g are real nilpotent orbits satisfying (O1)C = (O2)C, then either O1 = O2 orthese two orbits are incomparable in the closure order.

We begin with an overview of nilpotent orbits in semisimple real and complexLie algebras. In recognition of Theorem 46 (iii), and of the role played by the uniquemaximal complex nilpotent orbit Θreg(gC) throughout the article, Section 5.2.2 reviewsthe closure orders on the sets of real and complex nilpotent orbits. In Section 5.2.3, werecall some of the details underlying the use of decorated partitions to index nilpotentorbits.

Section 5.3 is devoted to the proof of Theorem 46. In Section 5.3.1, we representnilpotent orbit complexification as a poset map ϕg between the collections of real andcomplex nilpotent orbits. Next, we show this map to have a convenient description interms of decorated partitions. Section 5.3.2 then directly addresses the proof of Theo-rem 46 (i), formulated as a characterization of whenϕg is surjective. Using Proposition50, we reduce this exercise to one of characterizing surjectivity for g simple. Togetherwith the observation that surjectivity implies g is quasi-split and is implied by g beingsplit, Proposition 50 allows us to complete the proof of Theorem 46 (i).

We proceed to Section 5.3.3, which provides the proof of Theorem 46 (ii). Theessential ingredient is Kottwitz’s work [54]. We also include Proposition 55, whichgives an interesting sufficient condition for a complex nilpotent orbit to be in the imageof ϕg.

In Section 5.3.4, we give a proof of Theorem 46 (iii). Our proof makes extensive useof the Kostant-Sekiguchi Correspondence, the relevant parts of which are mentioned.

5.2 Background

5.2.1 Real and Complex Nilpotent Orbits

We begin by fixing some of the objects that will persist throughout this chapter. Letg be a finite-dimensional semisimple real Lie algebra with adjoint group G. Also, letgC := g⊗R C be the complexification of g, whose adjoint group is the complexificationGC. One has the adjoint representations

Ad : G→ GL(g) and AdC : GC → GL(gC)


of G and GC, respectively. Differentiation then gives the adjoint representations of gand gC, namely

ad : g → gl(g) and adC : gC → gl(gC).

Recall that an element ξ ∈ g (resp. ξ ∈ gC) is called nilpotent if ad(ξ) : g → g (resp.adC(ξ) : gC → gC) is a nilpotent vector space endomorphism. The nilpotent cone N(g)

(resp. N(gC)) is then the subvariety of nilpotent elements of g (resp. gC). A real (resp.complex) nilpotent orbit is an orbit of a nilpotent element in g (resp. gC) under the adjointrepresentation ofG (resp. GC). Since the adjoint representation occurs by means of Liealgebra automorphisms, a real (resp. complex) nilpotent orbit is equivalently definedto be a G-orbit (resp. GC-orbit) in N(g) (resp. N(gC)). By virtue of being an orbit of asmoothG-action, each real nilpotent orbit is an immersed submanifold of g. However,asGC is a complex linear algebraic group, a complex nilpotent orbit is a smooth locallyclosed complex subvariety of gC.

5.2.2 The Closure Orders

The sets N(g)/G and N(gC)/GC of real and complex nilpotent orbits are finite and carrythe so-called closure order. In both cases, this is a partial order defined by

O1 ≤ O2 if and only if O1 ⊆ O2. (5.1)

In the real case, one takes closures in the classical topology on g. For the complex case,note that a complex nilpotent orbit Θ is a constructible subset of gC, so that its Zariskiand classical closures agree. Accordingly, Θ shall denote this common closure.

Example 1. Suppose that gC = sln(C), whose adjoint group is GC = PSLn(C). Thenilpotent elements of sln(C) are precisely the nilpotent n × n matrices, so that thenilpotent PSLn(C)-orbits are exactly the (GLn(C)-) conjugacy classes of nilpotent ma-trices. The latter are indexed by the partitions of n via Jordan canonical forms. Givena partition λ = (λ1, λ2, . . . , λk) of n, let Θλ be the PSLn(C)-orbit of the nilpotent matrixwith Jordan blocks of sizes λ1, λ2, . . . , λk, read from top-to-bottom. It is a classical re-sult of Gerstenhaber [34] that Θλ ≤ Θµ if and only if λ ≤ µ in the dominance order.(See [70] for a precise definition of this order.)

The poset N(gC)/GC has a unique maximal element Θreg(gC), called the regularnilpotent orbit. It is the collection of all elements of gC which are simultaneouslyregular and nilpotent. In the framework of Example 1, Θreg(sln(C)) corresponds to thepartition (n).


5.2.3 Partitions of Nilpotent Orbits

Generalizing Example 1, it is often natural to associate a partition to each real and com-plex nilpotent orbit. One sometimes endows these partitions with certain decorationsand then uses decorated partitions to enumerate nilpotent orbits. It will be advanta-geous for us to recall the construction of the underlying (undecorated) partitions. Ourexposition will be largely based on Chapters 5 and 9 of [18].

Suppose that g comes equipped with a faithful representation g ⊆ gl(V) = EndF(V),where V is a finite-dimensional vector space over F = R or C.1 The choice of V deter-mines an assignment of partitions to nilpotent orbits in both g and gC. To this end, fixa real nilpotent orbit O ⊆ N(g) and choose a point ξ ∈ O. We may include ξ as thenilpositive element of an sl2(R)-triple (ξ, h, η), so that

[ξ, η] = h, [h, ξ] = 2ξ, [h, η] = −2η.

Regarding V as an sl2(R)-module, one has a decomposition into irreducibles,

V =

k⊕j=1

Vλj ,

where Vλj denotes the irreducible λj-dimensional representation of sl2(R) over F. Letus require that λ1 ≥ λ2 ≥ . . . ≥ λk, so that (λ1, λ2, . . . , λk) is a partition of dimF(V).Accordingly, we define the partition of O to be

λ(O) := (λ1, λ2, . . . , λk).

It can be established that λ(O) depends only on O.The faithful representation V of g canonically gives a faithful representation V of

gC. Indeed, if V is over C, then one has an inclusion gC ⊆ gl(V) (so V = V). If Vis over R, then the inclusion g ⊆ gl(V) complexifies to give a faithful representationgC ⊆ gl(VC) (ie. V = VC). In either case, one proceeds in analogy with the real nilpotentcase, using the faithful representation to yield a partition λ(Θ) of a complex nilpotentorbitΘ ⊆ N(gC). The only notable difference with the real case is that sl2(R) is replacedwith sl2(C).

Example 2. One can use the framework developed above to index the nilpotent orbitsin sln(C) using the partitions of n. This coincides with the indexing given in Example

1Since g is semisimple, the adjoint representation is a canonical choice of faithful V . Nevertheless, itwill be advantageous to allow for different choices.


1.

Example 3. The nilpotent orbits in sln(R) are indexed by the partitions of n, after onereplaces certain partitions with decorated counterparts. Indeed, if λ is a partition of nhaving only even parts, we replace λ with the decorated partitions λ+ and λ−. Other-wise, we leave λ undecorated.

Example 4. Suppose that n ≥ 3 and consider g = su(p, q) with 1 ≤ q ≤ p and p+q = n.This Lie algebra is a real form of sln(C). Now, let us regard a partition of n as a Youngdiagram with n boxes. Furthermore, recall that a signed Young diagram is a Youngdiagram whose boxes are marked with + or −, such that the signs alternate acrosseach row (for more details, see Chapter 9 of [18]). We restrict our attention to thesigned Young diagrams of signature (p, q), namely those for which + and − appearwith respective multiplicities p and q. It turns out that the nilpotent orbits in su(p, q)

are indexed by the signed Young diagrams of signature (p, q).

Example 5. Suppose that gC = so2n(C) with n ≥ 4. Taking our faithful representationto be C2n, nilpotent orbits in so2n(C) are assigned partitions of 2n. The partitionsrealized in this way are those in which each even part appears with even multiplicity.One extends these partitions to an indexing set by replacing each λ having only evenparts with the decorated partitions λ+ and λ−.

Example 6. Suppose that n ≥ 3 and consider g = so(p, q) with 1 ≤ q ≤ p and p + q =

n. Note that so(p, q) is a real form of gC = son(C). As with Example 4, we willidentify partitions of n with Young diagrams having n boxes. We begin with thesigned Young diagrams of signature (p, q) such that each even-length row appearswith even multiplicity and has its leftmost box marked with +. To obtain an indexingset for the nilpotent orbits in so(p, q), we decorate two classes of these signed Youngdiagrams Y. Accordingly, if Y has only even-length rows, then remove Y and addthe four decorated diagrams Y+,+, Y+,−, Y−,+, and Y−,−. Secondly, suppose that Y hasat least one odd-length row, and that each such row has an even number of boxesmarked +, or that each such row has an even number of boxes marked −. In this case,we remove Y and add the decorated diagrams Y+ and Y−.


5.3 Nilpotent Orbit Complexification

5.3.1 The Complexification Map

There is a natural way in which a real nilpotent orbit determines a complex one. In-deed, the inclusion N(g) ⊆ N(gC) gives rise to a map

ϕg : N(g)/G→ N(gC)/GC

O 7→ OC.

Concretely, OC is just the unique complex nilpotent orbit containing O, and we shallcall it the complexification of O. Let us then call ϕg the complexification map for g.

It will be prudent to note that the process of nilpotent orbit complexification iswell-behaved with respect to taking partitions. More explicitly, we have the followingproposition.

Proposition 47. Suppose that g is endowed with a faithful representation g ⊆ gl(V). If O isa real nilpotent orbit, then λ(OC) = λ(O).

Proof. Choose a point ξ ∈ O and include it in an sl2(R)-triple (ξ, h, η) as in Sec-tion 5.2.3. Note that (ξ, h, η) is then additionally an sl2(C)-triple in gC. Hence, wewill prove that the faithful representation V of gC decomposes into irreducible sl2(C)-representations according to the partition λ(O).

Let us write λ(O) = (λ1, . . . , λk), so that

V =

k⊕j=1

Vλj (5.2)

is the decomposition of V into irreducible sl2(R)-representations. If V is over C, thenV = V and (5.2) is a decomposition of V into irreducible sl2(C)-representations. If V isover R, then V = VC and

VC =

k⊕j=1

(Vλj)C

is the decomposition of V into irreducible representations of sl2(C). In each of thesetwo cases, we have λ(OC) = λ(O).

Proposition 47 allows us to describe ϕg in more combinatorial terms. To this end,fix a faithful representation g ⊆ gl(V). As in Examples 2–6, we obtain index sets I(g)


and I(gC) of decorated partitions for the real and complex nilpotent orbits, respec-tively. We may therefore regard ϕg as a map

ϕg : I(g) → I(gC).

Now, let P(gC) be the set of all partitions of the form λ(Θ), with Θ ⊆ gC a complexnilpotent orbit. One has the map

I(gC) → P(gC),

sending a decorated partition to its underlying partition. Proposition 47 is then thestatement that the composite map

I(g)ϕg−→ I(gC) → P(gC)

sends an index in I(g) to its underlying partition. Let us denote this composite mapby ψg : I(g) → P(gC).

We will later give a characterization of those semisimple real Lie algebras g forwhich ϕg is surjective. To help motivate this, we investigate the matter of surjectivityin some concrete examples.

Example 7. Recall the parametrizations of nilpotent orbits in g = sln(R) and gC =

sln(C) outlined in Examples 3 and 2, respectively. We see that I(gC) = P(gC) andϕg = ψg. The surjectivity of ϕg then follows immediately from that of ψg.

Example 8. Let the nilpotent orbits in g = su(n,n) be parametrized as in Example 4.We then have gC = sl2n(C), whose nilpotent orbits are indexed by the partitions of2n. Given such a partition λ, let Y denote the corresponding Young diagram. Since Yhas an even number of boxes, it has an even number, 2k, of odd-length rows. Labelthe leftmost box in k of these rows with +, and label the leftmost box in each of theremaining k rows with −. Now, complete this labelling to obtain a signed Youngdiagram Y, noting that Y then has signature (n,n). Hence, Y corresponds to a nilpotentorbit in su(n,n) and ψg(Y) = λ. It follows that ψg is surjective. Since I(gC) = P(gC)

and ϕg = ψg, we have shown ϕg to be surjective. A similar argument establishessurjectivity when g = su(n+ 1, n).

Example 9. Let us consider g = so(2n + 2, 2n), with nilpotent orbits indexed as inExample 6. Noting Example 5, a partition λ of 4n + 2 represents a nilpotent orbit ingC = so4n+2(C) if and only if each even part of λ occurs with even multiplicity. Since4n+2 is even and not divisible by 4, it follows that any such λ has exactly 2k odd parts


for some k ≥ 1. Let Y be the Young diagram corresponding to λ, and label the leftmostbox in k − 1 of the odd-length rows with +. Next, label the leftmost box in each ofk− 1 different odd-length rows with −. Finally, use + to label the leftmost box in eachof the two remaining odd-length rows. Let Y be any completion of our labelling to asigned Young diagram, such that the leftmost box in each even-length row is markedwith +. Note that Y has signature (2n+ 2, 2n). It follows that Y represents a nilpotentorbit in so(2n+2, 2n) andψg(Y) = λ. Furthermore, I(gC) = P(gC) andϕg = ψg, so thatϕg is surjective.

Example 10. Suppose that g = so(2n+1, 2n−1), whose nilpotent orbits are parametrizedin Example 6. Let the nilpotent orbits in gC = so4n(C) be indexed as in Example 5.There exist partitions of 4n having only even parts, with each part appearing an evennumber of times. Let λ be one such partition, which by Example 6 represents a nilpo-tent orbit in so4n(C). Note that every signed Young diagram with underlying partitionλ must have signature (2n, 2n). In particular, λ cannot be realized as the image underψg of a signed Young diagram indexing a nilpotent orbit in so(2n+1, 2n−1). It followsthat ψg and ϕg are not surjective.

5.3.2 Surjectivity

We now address the matter of classifying those semisimple real Lie algebras g forwhich ϕg is surjective. To proceed, we will require some additional machinery. Letp ⊆ g be the (−1)-eigenspace of a Cartan involution, and let a be a maximal abeliansubspace of p. Also, let h be a Cartan subalgebra of g containing a, and choose afundamental Weyl chamber C ⊆ h. Given a complex nilpotent orbit Θ ⊆ gC, thereexists an sl2(C)-triple (ξ, h, η) in gC with the property that ξ ∈ Θ and h ∈ C. Theelement h ∈ C is uniquely determined by this property, and is called the characteristicof Θ. Theorem 1 of [26] then states that Θ ∩ g 6= ∅ if and only if h ∈ a. If g is split, thena = h, and the following lemma is immediate.

Lemma 48. If g is split, then ϕg is surjective.

Let us now consider necessary conditions for surjectivity. To this end, recall that g iscalled quasi-split if there exists a subalgebra b ⊆ g such that bC is a Borel subalgebra ofgC. However, the following characterization of being quasi-split will be more suitablefor our purposes.

Lemma 49. The Lie algebra g is quasi-split if and only if Θreg(gC) is in the image of ϕg. Inparticular, g being quasi-split is a necessary condition for ϕg to be surjective.


Proof. Proposition 5.1 of [61] states that g is quasi-split if and only if g contains a regu-lar nilpotent element of gC. Since Θreg(gC) consists of all such elements, this is equiva-lent to havingΘreg(gC)∩g 6= ∅ hold. This latter condition holds precisely whenΘreg(gC)

is in the image of ϕg.

Lemmas 48 and 49 establish that ϕg being surjective is a weaker condition thanhaving g be split, but stronger than having g be quasi-split. Furthermore, since su(n,n)is not a split real form of sl2n(C), Example 8 establishes that surjectivity is strictlyweaker than g being split. Yet, as so(2n+1, 2n−1) is a quasi-split real form of so4n(C),Example 10 demonstrates that surjectivity is strictly stronger than having g be quasi-split. To obtain a more precise measure of the strength of the surjectivity condition,we will require the following proposition.

Proposition 50. Suppose that g decomposes as a Lie algebra into

g =

k⊕j=1

gj,

where g1, . . . , gk are simple real Lie algebras. Let G1, . . . , Gk denote the respective adjointgroups.

(i) The mapϕg : N(g)/G→ N(gC)/GC is surjective if and only each orbit complexificationmap ϕgj : N(gj)/Gj → N((gj)C)/(Gj)C is surjective.

(ii) The Lie algebra g is quasi-split if and only if each summand gj is quasi-split.

Proof. For each j ∈ 1, . . . , k, let πj : g → gj be the projection map. Note that ξ ∈ g isnilpotent if and only if πj(ξ) is nilpotent in gj for each j. It follows that

π : N(g) → k∏j=1

N(gj)

ξ 7→ (πj(ξ))kj=1

defines an isomorphism of real varieties. Note that G =∏k

j=1Gj, with the formergroup acting on N(g) and the latter group acting on the product of nilpotent cones.

One then sees that π is G-equivariant, so that it descends to a bijection

π : N(g)/G→ k∏j=1

N(gj)/Gj.


Analogous considerations give a second bijection

π : N(gC)/GC → k∏j=1

N((gj)C)/(Gj)C.

Furthermore, we have the commutative diagram

N(g)/G

ϕg

π //∏k

j=1N(gj)/Gj∏kj=1 ϕgj

N(gC)/GCπC //

∏kj=1N((gj)C)/(Gj)C

. (5.3)

Hence, ϕg is surjective if and only if∏k

j=1ϕgj is so, proving (i).By Lemma 49, proving (ii) will be equivalent to proving thatΘreg(gC) is in the image

of ϕg if and only if Θreg(gj) is in the image of ϕgj for all j. Using the diagram (5.3), thiswill follow from our proving that the image of Θreg(gC) under π is the k-tuple of theregular nilpotent orbits in the (gj)C, namely that

π(Θreg(gC)) = (Θreg(((gj)C)))kj=1. (5.4)

To see this, note that∏k

j=1Θreg(((gj)C)) is the GC =∏k

j=1(Gj)C-orbit of maximal di-mension in

∏kj=1N(((gC)j)). This orbit is therefore the image of Θreg(gC) under the

GC-equivariant variety isomorphism N(gC) ∼=∏k

j=1N(((gj)C)), implying that (5.4)holds.

In light of Proposition 50, we address ourselves to classifying the simple real Liealgebras g with surjective orbit complexification maps ϕg. Noting Lemma 49, we mayassume g to be quasi-split. Since g being split is a sufficient condition for surjectivity,we are further reduced to finding those quasi-split simple g which are non-split buthave surjectiveϕg. It follows that g belongs to one of the four families su(n,n), su(n+1, n), so(2n+2, 2n), and so(2n+1, 2n−1), or that g = EII, the non-split, quasi-split realform of E6 (see Appendix C3 of [49]). Our examples establish that ϕg is surjective forg = su(n,n), g = su(n+ 1, n), and g = so(2n+ 2, 2n), while Example 10 demonstratesthat surjectivity does not hold for g = so(2n + 1, 2n − 1). Also, a brief examinationof the computations in [27] reveals that ϕg is surjective for g = EII. We then have thefollowing characterization of the surjectivity condition.

Theorem 51. If g is a semisimple real Lie algebra, then ϕg is surjective if and only if g isquasi-split and has no simple summand of the form so(2n+ 1, 2n− 1).


Proof. If ϕg is surjective, then Lemma 49 implies that g is quasi-split. Also, Proposi-tion 50 implies that each simple summand of g has a surjective orbit complexificationmap, and the above discussion then establishes that g has no simple summand of theform so(2n + 1, 2n − 1). Conversely, assume that g is quasi-split and has no simplesummand of the form so(2n+ 1, 2n− 1). By Proposition 50 (ii), each simple summandof g is quasi-split. Furthermore, the above discussion implies that the only quasi-splitsimple real Lie algebras with non-surjective orbit complexification maps are those ofthe form s0(2n + 1, 2n − 1). Hence, each simple summand of g has a surjective orbitcomplexification map, and Proposition 50 (i) implies that ϕg is surjective.

5.3.3 The Image of ϕg

Having investigated the surjectivity of ϕg, let us consider the more subtle matter ofcharacterizing its image. Accordingly, let σg : gC → gC denote complex conjugationwith respect to the real form g ⊆ gC. The following lemma will be useful.

Lemma 52. If Θ ⊆ gC is a complex nilpotent orbit, then so is σg(Θ).

Proof. Note that σg integrates to a real Lie group automorphism

τ : Gsc → Gsc,

where (G)sc is the connected, simply-connected Lie group with Lie algebra gC. If g ∈Gsc and ξ ∈ gC, then

σg(AdC(g)(ξ)) = AdC(τ(g))(σg(ξ)).

Hence, σg sends the Gsc-orbit of ξ to the Gsc-orbit of σg(ξ). To complete the proof,we need only observe that Gsc-orbits coincide with GC-orbits in gC, and that σg(ξ) isnilpotent whenever ξ is nilpotent.

We may now use σg to explicitly describe the image of ϕg when g is quasi-split.

Theorem 53. If Θ is a complex nilpotent orbit, the condition σg(Θ) = Θ is necessary for Θ tobe in the image of ϕg. If g is quasi-split, then this condition is also sufficient.

Proof. Assume that Θ belongs to the image of ϕg, so that there exists ξ ∈ Θ ∩ g. Notethat σg(Θ) is then the complex nilpotent orbit containing σg(ξ) = ξ, meaning thatσg(Θ) = Θ. Conversely, assume that g is quasi-split and that σg(Θ) = Θ. The lat-ter means precisely that Θ is defined over R with respect to the real structure on gC

induced by the inclusion g ⊆ gC. Theorem 4.2 of [54] then implies that Θ ∩ g 6= ∅.


Using Theorem 53, we will give an interesting sufficient condition for a complexnilpotent orbit to be in the image ofϕg when g is quasi-split. In order to proceed, how-ever, we will need a better understanding of the way in which σg permutes complexnilpotent orbits. To this end, we have the following lemma.

Lemma 54. Suppose that g comes with the faithful representation g ⊆ gl(V), where V is overR. If Θ is a complex nilpotent orbit, then λ(σg(Θ)) = λ(Θ).

Proof. Choose an sl2(C)-triple (ξ, h, η) in gC with ξ ∈ Θ. Since σg preserves Lie brack-ets, it follows that (σg(ξ), σg(h), σg(η)) is also an sl2(C)-triple. The exercise is then toshow that our two sl2(C)-triples give isomorphic representations of sl2(C) on V = VC.For this, it will suffice to prove that h and σg(h) act on VC with the same eigenvalues,and that their respective eigenspaces for a given eigenvalue are equi-dimensional. Tothis end, let σV : VC → VC be complex conjugation with respect to V ⊆ VC. Note that

σg(h) · (σV(x)) = σV(h · x)

for all x ∈ VC, where · is used to denote the action of gC on VC. Hence, if x is aneigenvector of h with eigenvalue λ ∈ R, then σV(x) is an eigenvector of σg(h) witheigenvalue λ. We conclude that h and σg(h) have the same eigenvalues. Furthermore,their respective eigenspaces for a fixed eigenvalue are related by σV , and so are equi-dimensional.

We now have the following

Proposition 55. Let g be a quasi-split semisimple real Lie algebra endowed with a faithfulrepresentation g ⊆ gl(V), where V is over R. If Θ is the unique complex nilpotent orbit withpartition λ(Θ), then Θ is in the image of ϕg.

Proof. By Lemma 54, σg(Θ) is a complex nilpotent orbit with partition λ(Θ), and ourhypothesis on Θ gives σg(Θ) = Θ. Theorem 53 then implies that Θ is in the image ofϕg.

A few remarks are in order.

Remark. One can use Proposition 55 to investigate whetherϕg is surjective without ap-pealing to the partition-type description of ϕg discussed in Section 5.3.1. For instance,suppose that g = so(2n + 2, 2n), a quasi-split real form of gC = so4n+2(C). We referthe reader to Example 5 for the precise assignment of partitions to nilpotent orbits inso4n+2(C). In particular, note that a complex nilpotent orbit is the unique one withits partition if and only if the partition does not have all even parts. Furthermore, as


discussed in Example 9, there do not exist partitions of 4n+ 2 having only even partssuch that each part appears with even multiplicity. Hence, each complex nilpotentorbit is specified by its partition, so Proposition 55 implies that ϕg is surjective.

Remark. The converse of Proposition 55 does not hold. Indeed, suppose that g =

so(2n, 2n), the split real form of gC = so4n(C). Recalling Example 5, every partition of4nwith only even parts, each appearing with even multiplicity, is the partition of twodistinct complex nilpotent orbits. Yet, Lemma 48 implies that ϕg is surjective, so thatthese orbits are in the image of ϕg.

5.3.4 Fibres

In this section, we investigate the fibres of the orbit complexification mapϕg : N(g)/G→N(gC)/GC. In order to proceed, it will be necessary to recall some aspects of theKostant-Sekiguchi Correspondence. To this end, fix a Cartan involution θ : g → g.Letting k and p denote the 1 and (−1)-eigenspaces of θ, respectively, we obtain theinternal direct sum decomposition

g = k⊕ p.

This gives a second decomposition

gC = kC ⊕ pC.

Let K ⊆ G and KC ⊆ GC be the connected closed subgroups with respective Lie al-gebras k and kC. The Kostant-Sekiguchi Correspondence is one between the nilpotentorbits in g and the KC-orbits in the (KC-invariant) subvariety pC ∩N(gC) of gC.

Theorem 56 (The Kostant-Sekiguchi Correspondence). There is a bijective correspon-dence

N(g)/G→ (pC ∩N(gC))/KC

O 7→ O∨

with the following properties.

(i) It is an isomorphism of posets, where (pC∩N(gC))/KC is endowed with the closure order(5.1).

(ii) If O is a real nilpotent orbit, then O and O∨ are K-equivariantly diffeomorphic.


The first property was established by Barbasch and Sepanski in [5], while the sec-ond was proved by Vergne in [74]. Each paper makes extensive use of Kronheimer’sdesciption of nilpotent orbits from [55].

We now prove two preliminary results, the first of which is a direct consequenceof the Kostant-Sekiguchi Correspondence.

Lemma 57. If O is a real nilpotent orbit, then O is the unique G-orbit of maximal dimensionin O.

Proof. Suppose that O ′ 6= O is another G-orbit lying in O. By Property (i) in Theorem56, it follows that (O ′)∨ is an orbit in (O∨) different from O∨. However, O∨ is an orbit ofthe complex algebraic group KC under an algebraic action, and therefore is the uniqueorbit of maximal dimension in its closure. Hence, dimR((O

′)∨) < dimR(O∨). Property

(ii) of Theorem 56 implies that the Kostant-Sekiguchi Correspondence preserves realdimensions, so that dimR(O

′) < dimR(O).

We will also require some understanding of the relationship between theG-stabilizerof ξ ∈ g and the GC-stabilizer of ξ, viewed as an element of gC. Denoting these stabi-lizers by Gξ and (GC)ξ, respectively, we have the following lemma.

Lemma 58. If ξ ∈ g, then Gξ is a real form of (GC)ξ.

Proof. We are claiming that the Lie algebra of (G)ξ is the complexification of the Liealgebra of Gξ. The former is (gC)ξ = η ∈ gC : [η, ξ] = 0, while the Lie algebra ofGξ is gξ = η ∈ g : [η, ξ] = 0. If η = η1 + iη2 ∈ gC with η1, η2 ∈ g, then [η, ξ] =

[η1, ξ] + i[η2, ξ]. So, η ∈ (gC)ξ if and only if η1, η2 ∈ gξ. This is equivalent to thecondition that η ∈ ((g)ξ)C ⊆ gC, so that (gC)ξ = (gξ)C.

We may now prove the main result of this section.

Theorem 59. If O1 and O2 are real nilpotent orbits with the property that (O1)C = (O2)C,then either O1 = O2 or O1 and O2 are incomparable in the closure order. In other words, eachfibre of ϕg consists of pairwise incomparable nilpotent orbits.

Proof. Assume that O1 and O2 are comparable. Without the loss of generality, O1 ⊆ O2.We will prove that O1 = O2, which by Lemma 57 will amount to showing that thedimensions of O1 and O2 agree. To this end, choose points ξ1 ∈ O1 and ξ2 ∈ O2. Since(O1)C = (O2)C, we have dimC((GC)ξ1) = dimC((GC)ξ2). Using Lemma 58, this becomesdimR(Gξ1) = dimR(Gξ2). Hence, the (real) dimensions of O1 and O2 coincide.

Chapter 6

Equivariant Contact Geometry and theLeBrun-Salamon Conjecture

In the following chapter, we study G-equivariant contact geometry in the context ofLeBrun and Salamon’s conjectural classification of Fano contact varieties. This is basedon joint work with Steven Rayan, and the exposition follows the manuscript [21].While Steven and I were both fully participant in all aspects of this project, I gen-erally contributed the Lie-theoretic techniques while Steven provided the geometricones.


Fano varieties with complex contact structures have been studied enthusiasticallyover the last half century, in large part due to their distinguished position at the in-tersection of complex algebraic geometry and real differential geometry. A compactquaternionic Kahler manifold with positive curvature always supports an S2-bundle— its twistor space — the total space of which is a Fano contact manifold. As presentedin [48], the LeBrun-Salamon conjecture [56] posits that every Fano contact manifoldwith b2 = 1 arises as the projectivization of the minimal nilpotent orbit of some simplealgebraic group. If the conjecture were true, then every compact quaternionic Kahlervariety with positive curvature would necessarily be homogeneous, and so progresson the LeBrun-Salamon conjecture is crucial to resolving an outstanding geometricclassification problem within Riemannian geometry. The work of Beauville [6] is thestrongest evidence thus far for the validity of the conjecture. For multiple points ofview on Fano contact varieties, including the minimal rational curves and Mori the-

82

CHAPTER 6. EQUIVARIANT CONTACT GEOMETRY AND THE LEBRUN-SALAMON CONJECTURE83

ory approaches, we refer the reader to [16, 44, 47, 58, 59].

On another front, complex contact manifolds have been studied in the context ofequivariant geometry. Most notably, Boothby [11, 12] gives a complete classificationof those compact simply-connected complex contact manifolds which are acted upontransitively by their respective groups of contact automorphisms (the so-called homo-geneous complex contact manifolds). He identifies each with the projectivization of asuitable minimal nilpotent orbit.

This article presents some results on equivariant contact geometry for partial flagvarieties. Firstly, we give a self-contained proof of the following special case of theLeBrun-Salamon conjecture.

Theorem 60. Assume that G is of type ADE, and let X be a partial flag variety of G withb2(X) = 1. If X is endowed with a G-invariant complex contact structure, then there exists aG-equivariant isomorphism X ∼= P(Omin) of contact varieties.

While Theorem 60 is deducible from Boothby’s work, our argument differs sig-nificantly from that offered in [11, 12]. Our approach is instead based on a sequenceof results concerning the geometry (both equivariant and non-equivariant) of partialflag varieties G/P, where P is a parabolic subgroup. Specifically, we prove that a G-invariant corank-1 subbundle E of TG/P is completely determined as such by the iso-morphism class of the quotient line bundle TG/P/E (see Proposition 62). This leads usto prove Proposition 64, which describes the contact line bundle of a G-invariant con-tact structure onG/P in terms of the isomorphism between Pic(G/P) and the group of1-dimensional P-representations. Proposition 66 and Theorem 68 then combine to giveus the desired G-equivariant contact variety isomorphism between G/P and P(Omin).

Secondly, we offer a detailed description of the contact manifold in Theorem 60when G is of type Dn. This manifold is precisely the Grassmannian GrB(2,C2n) ofthose 2-planes in C2n which are isotropic with respect to the complex-bilinear dotproduct. While there are descriptions of the SO2n(C)-invariant contact distributionE onGrB(2,C2n) appearing in the literature (e.g. [44]), ours is global and canonical. In-deed, we use the classical identification of the tangent bundle of the full GrassmannianGr(2,C2n) with Hom(F,O⊕2n/F), where F is the tautological bundle onGr(2,C2n). Wethen present E explicitly as a subbundle of the pullback toGrB(2,C2n) of Hom(F,O⊕2n/F).


6.2 Review of Properties of Fano Contact Varieties

Here, we review the salient features of complex contact varieties in general and Fanocontact varieties in particular. Let X be a smooth complex variety of complex dimen-sion 2n + 1 for some n ≥ 0, and let ι : E → TX be a rank-2n holomorphic subbundleof the tangent bundle TX. We say that the pair (X,E) is contact if, in the short exactsequence

0 −→ Eι−→ TX

θ−→ L −→ 0 (6.1)

induced by ι, the composition of the Lie bracket on sections of TX with the quotientmap θ is an L-twisted bilinear form that is non-degenerate along E. In keeping withthe literature, we call the subbundle E the contact distribution and the quotient L thecontact line bundle of (X,E). If there exists an E → X for which the pair (X,E) is contact,we say that X admits a contact structure.

From now on, we also assume that X is projective, that b2(X) = 1, and that X admitsa contact structure, the distribution of which is E. Let L be the associated contactline bundle. We use KX and K∨

X , respectively, for the canonical and anticanonicalline bundles of X. In this case, X is Fano with Pic(X) ∼= Z and K∨

X∼= L⊗n+1. This

characterization is a consequence of a theorem of Demailly (Cor. 3 in [24]), applied toan earlier result of Kebekus, Peternell, Sommese, Wisniewski (Thm. 1.1 in [48]).

There are two possibilities: either the contact line bundle L is a generator of Pic(X)or it is not. If it is not, then L is a holomorphic (n + 1)-th root of K∨

X and L itselfhas nontrivial roots (namely, a generator of Pic(X)). In this case, X must be PN forsome N, by the well-known Kobayashi-Ochiai characterization of complex projectivespace [50]. Hence, whenever X is a projective Fano contact variety with b2 = 1 that isnot a projective space, then it must be that Pic(X) = Z · [L].

Taking these observations together, we have that:

• L is ample (in particular, it is the ample generator of Pic(X) whenever X PN),and

• if L ′ is any other contact line bundle on X, then there must exist a vector bundleisomorphism L ∼= L ′.

The second fact is true because Pic(X) ∼= Z and L and L ′ are holomorphic roots ofthe same line bundle (and hence degL = degL ′).


6.3 Partial Flag Varieties and Contact Structures

6.3.1 Basic Setup

Now we specialize to the case where X is a partial flag variety, namely X = G/PS forsome S ⊆ Π. In particular, X is necessarily Fano (see, for instance, Thm. V.1.4 in [51]).

With a view to eventually using the isomorphism (2.18), we will need a particularZ-basis of the group X∗(T)WS . For β ∈ Π, let hβ ∈ [gβ, g−β] be the correspondingsimple coroot. Note that the hβ form a basis of t dual to the basis of fundamentalweightsωβ ∈ X∗(T), β ∈ Π.

Lemma 61. The group X∗(T)WS has a Z-basis of ωβ : β ∈ Π \ S.

Proof. Note that δ ∈ X∗(T) belongs to X∗(T)WS if and only if δ is fixed by each simplereflection sβ, β ∈ S. This holds if and only if δ is orthogonal to each simple root in S.The desired conclusion then follows from the fact that

δ =∑β∈Π

δ(hβ)ωβ =∑β∈Π

2(δ, β)

(β,β)ωβ

is the expression of δ as a linear combination of the fundamental weights.

We conclude this section with a proposition that will be of use later.

Proposition 62. If F1 and F2 areG-invariant corank-1 subbundles of TG/PS and the quotientsTG/PS/F1 and TG/PS/F2 are isomorphic as line bundles, then F1 = F2.

Proof. To begin, note that(TG/PS

)[e]

is canonically isomorphic to g/pS as a PS-representation,so that

TG/PS∼= G×PS (g/pS). (6.2)

The isomorphism (6.2) restricts to isomorphisms

F1 ∼= G×PS V1

andF2 ∼= G×PS V2,

whereV1 andV2 are codimension-1 PS-subrepresentations of g/pS. Since each T -weightspace of g/pS is 1-dimensional, each of V1 and V2 is obtained by removing a single


weight space from g/pS. Let γ1, γ2 ∈ ∆− be the weights discarded to obtain V1 and V2,respectively. We then have bundle isomorphisms

TG/PS/F1∼= G×PS ((g/pS)/V1) ∼= L(γ1)

andTG/PS/F2

∼= G×PS ((g/pS)/V2) ∼= L(γ2).

In particular, L(γ1) ∼= L(γ2) as line bundles overG/PS, so that γ1 = γ2. Hence, V1 = V2,implying that F1 and F2 identify with the same subbundle of G×PS (g/pS) under (6.2).This completes the proof.

In the case that F1 and F2 define contact structures, we have the following imme-diate

Corollary 63. If each of F1 and F2 is the distribution of a G-invariant contact structure onG/PS and Pic(G/PS) ∼= Z, then F1 = F2.

This is simply the result of combining Proposition 62 with the fact G/PS is Fano(and then applying the second observation listed at the end of Section 6.2).

6.3.2 The Projectivization of the Minimal Nilpotent Orbit

The material in 6.3.1 facilitates a worthwhile discussion of P(Omin) and its G-invariantcontact structure. To this end, suppose that ξ ∈ gλ \ 0, which determines a class [ξ] ∈P(Omin). By Proposition 9, theG-stabilizer of [ξ] is the standard parabolic subgroup PΛ,whereΛ is the collection of those simple roots which are orthogonal to λ. We thereforehave the G-variety isomorphism

ϕ : G/PΛ∼=−→ P(Omin) (6.3)

[g] 7→ [Ad(g)(ξ)].

It turns out that P(Omin) carries a distinguishedG-invariant contact structure, Emin ⊆TP(Omin). To obtain it, note that the Killing form on g restricts to a G-equivariant varietyisomorphism between Omin and a coadjoint orbit in g∗. The latter has the Kirillov-Kostant-Souriau symplectic structure, so that Omin is symplectic. The symplectic formon Omin has weight 1 with respect to the scaling action of C∗, and Lemma 1.4 of [6]then gives the desired contact structure on P(Omin).


Using Remark 2.3 from [6], one can more explicitly describe the bundle Emin. Let[ξ] ∈ P(Omin) be the class of a highest root vector, as above. Via the isomorphism (6.3),the fibre (Emin)[ξ] identifies with a codimension-1 subspace of g/pΛ, the tangent spaceof G/PΛ at the identity coset. Now, note that pΛ ⊆ (gλ)

⊥, the orthogonal complementof gλ with respect to the Killing form. Our fibre is then given by

(Emin)[ξ] = (gλ)⊥/pΛ. (6.4)

Since Emin is a G-invariant subbundle of TP(Omin), (6.4) can be used to determine thefibre of Emin over any point.

6.3.3 Reduction to the Case of a Maximal Parabolic

Let us begin to directly address the classification of partial flag varieties admittingG-invariant contact structures. To this end, assume S ⊆ Π is such that G/PS admitsa G-invariant contact structure E ⊆ TG/PS . In light of earlier remarks, we shall alsoassume that b2(G/PS) = 1. This second assumption imposes a significant constrainton the subsets S under consideration. Indeed, one has the opposite Schubert cell de-composition (2.6), so thatH2(G/PS;Z) is free of rank equal to the number of (complex)codimension-1 opposite Schubert cells. Since the codimension of B−wPS/PS in G/PS isthe length of a minimal-length coset representative in [w] ∈W/WS, the codimension-1opposite Schubert cells are those of the form B−sβPS/PS, β ∈ Π \ S. Hence, the con-dition b2(G/PS) = 1 implies that Π \ S has cardinality 1, so that S = Π \ α for someunique α ∈ Π. In other words, PS is a maximal parabolic subgroup of G.

6.3.4 The Contact Line Bundle on G/PS

We now give a more explicit description of the G-invariant contact structure E. Usingthe bundle isomorphism (6.2), we will regard the fibre E[e] as a codimension-1 PS-subrepresentation of g/pS. Of course, since E is a G-invariant subbundle of TG/PS , wealso have

E ∼= G×PS E[e]. (6.5)

Hence, the contact line bundle L = TG/PS/E is given by

L ∼= G×PS((g/pS)/E[e]

). (6.6)

Using (6.6), we can present L in terms of the description of equivariant line bundles


from 2.4.3. To The reader will want to recall the isomorphism (2.16) between Pic(G/PS)and X∗(T)WS

Proposition 64. If g is simply-laced, then the highest root λ belongs to X∗(T)WS and L isisomorphic to L(−λ).

Proof. We begin with two observations. Firstly, we have L ∼= L(γ) for some γ ∈X∗(T)WS . Secondly, since g is simply-laced, −λ is the unique anti-dominant root. Prov-ing the proposition will therefore amount to showing that γ ∈ ∆, and that γ is anti-dominant. For the former, note that each T -weight of g/pS is a root. It follows that theweight of the quotient representation (g/pS)/E[e] is also a root. Also, the isomorphisms(6.6) and L ∼= L(γ) together imply that γ is a weight of g/pS, so that γmust be a root.

To prove that γ is anti-dominant, we note that

L((n+ 1)γ) ∼= L⊗(n+1) ∼= K∨G/PS

, (6.7)

where 2n+ 1 is the (complex) dimension of G/PS. Also, the isomorphism (6.2) yields

K∨G/PS

= ∧2n+1TG/PS∼= G×PS (∧2n+1(g/pS)).

Letting µS be the weight of ∧2n+1(g/pS), we have

K∨G/PS

∼= L(µS). (6.8)

Combining (6.7) and (6.8), we conclude that (n+ 1)γ = µS. Since µS is anti-dominant,this implies that γ is anti-dominant.

Before proceeding to the next section, we note that our arguments allow us toquickly recover the following well-known fact.

Corollary 65. The subbundle Emin ⊆ TP(Omin) is the unique G-invariant contact structure onP(Omin).

Proof. Suppose that F ⊆ TP(Omin)is aG-invariant contact structure. Let us first assumeGto be of typeADE (so that g is simply-laced). Note that both Emin and F pull-back toG-invariant contact structures on G/PΛ under the isomorphism (6.3). By Proposition 64,both contact line bundles TP(Omin)/Emin and TP(Omin)/F pull-back to L(−λ) under (6.3).In particular, these bundles are isomorphic, and Proposition 62 then implies F = Emin.

If G is not of type ADE, then Pic(P(Omin)) ∼= Z. Now, it follows from Corollary 63that F = Emin.


6.4 A Classification ofG-Invariant Contact Structures on

G/P

6.4.1 The Main Theorem

We now consolidate the results presented in Sections 6.3.3 and 6.3.4. In light of Propo-sition 64, we will assume G to be of type ADE for the duration of this article. We thenhave the following relationship between the simple root α from 6.3.3 and the highestroot λ.

Proposition 66. The root α is the unique simple root not orthogonal to λ.

Proof. By Lemma 61, X∗(T)WS is freely generated by ωα. Since λ ∈ X∗(T)WS by Propo-sition 64, it follows that λ = kωα for some non-zero k ∈ Z. Also, we may write

kωα = λ =∑β∈Π

λ(hβ)ωβ =∑β∈Π

2〈λ, β〉〈β,β〉

ωβ.

Hence, for β ∈ Π, we have 〈λ, β〉 = 0 if and only if β 6= α.

Before continuing, we note the following implication of Proposition 66 for partialflag varieties in type A.

Corollary 67. Suppose that G = SLn(C) with n ≥ 3. There does not exist a partial flag va-riety X of SLn(C) with b2(X) = 1 admitting an SLn(C)-invariant contact structure. Equiva-lently, none of the Grassmannians Gr(k,Cn), 1 ≤ k ≤ n − 1, supports an SLn(C)-invariantcontact structure.

Proof. By Proposition 66, the existence of such an X would imply that there was aunique simple root not orthogonal to the highest root λ. However, for G = SLn(C),n ≥ 3, there are exactly two simple roots not orthogonal to λ. The formulation interms of Grassmannians follows from their being the partial flag varieties of SLn(C)having b2 = 1.

Remark. Corollary 67 has an interesting consequence when n is an even positive in-teger. Indeed, the odd-dimensional projective space Pn−1 is then isomorphic to theprojectivization of the minimal nilpotent orbit of Sp

n(C). In particular, Pn−1 admits an

Spn(C)-invariant contact structure. Yet, Corollary 67 implies that this contact structure

is not SLn(C)-invariant for n ≥ 4.


Let us return to the matter at hand. Proposition 66 establishes that S = Π \ α isthe collection of those simple roots which are orthogonal to λ, namely S = Λ. Hence,G/PS = G/PΛ, which isG-equivariantly isomorphic to P(Omin) via (6.3). It therefore re-mains to prove that (6.3) is additionally an isomorphism of contact varieties, recallingthat G/PS = G/PΛ has the G-invariant contact structure E ⊆ TG/PΛ fixed in 6.3.3.

Theorem 68. The map ϕ : G/PΛ → P(Omin) in (6.3) is an isomorphism of contact varieties.

Proof. We are claiming that ϕ∗(Emin) coincides with E when the former is regarded asa subbundle of TG/PΛ . Observing that each of ϕ∗(Emin) and E is a G-invariant corank-1subbundle of TG/PΛ , Proposition 62 allows us to reduce this to showing TG/PΛ/ϕ

∗(Emin)

and TG/PΛ/E to be isomorphic as line bundles. The second line bundle is isomor-phic to L(−λ) by Proposition 64, so we are further reduced to proving that the fibre(TG/PΛ/ϕ

∗(Emin))[e] has weight −λ as a T -representation.Let d[e]ϕ : (TG/PΛ)[e] → (TP(Omin))[ξ] (where [ξ] = ϕ([e])) be the differential of ϕ at

[e]. Since ϕ is T -equivariant, d[e]ϕ is an isomorphism of T -representations. Further-more, d[e]ϕ(ϕ∗(Emin)[e]) = (Emin)[ξ], so that (TG/PΛ/ϕ

∗(Emin))[e] and (TP(Omin))[ξ]/(Emin)[ξ]

are isomorphic T -representations. We also have an isomorphism (TP(Omin))[ξ]∼= g/pΛ

from Section 6.3.2, under which (Emin)[ξ] identifies with (gλ)⊥/pΛ. Putting everything

together, we have

(TG/PΛ/ϕ∗(Emin))[e] ∼= (TP(Omin))[ξ]/(Emin)[ξ] ∼= g/(gλ)

⊥. (6.9)

Since we have(gλ)

⊥ = b⊕⊕

β∈∆−\−λ

gβ,

(6.9) implies that (TG/PΛ/ϕ∗(Emin))[e] is indeed the 1-dimensional T -representation of

weight −λ.

6.4.2 Example: The Grassmannian of Isotropic 2-Planes in C2n

We now describe a class of explicit examples that satisfy the hypotheses of Theorem60. To this end, let us set G = SO2n(C) with n ≥ 4. 1 Given θ ∈ R/(2πZ), consider the2× 2matrix

R(θ) :=

[cos(θ) − sin(θ)sin(θ) cos(θ)

].

1Having developed this chapter forG simply-connected, one really should replace SO2n(C) with itssimply-connected double-cover Spin2n(C). However, the results of this section are readily seen to holdwhen G = SO2n(C).


For θ1, θ2, . . . , θn ∈ R/(2πZ), we define R(θ1, θ2, . . . , θn) to be the 2n × 2n block-diagonal matrix R(θ1) ⊕ R(θ2) ⊕ · · · ⊕ R(θn). Note that the R(θ1, θ2, . . . , θn) constitutea maximal torus of the compact real form SO(2n) ⊆ SO2n(C). Let T ⊆ SO2n(C) be thecomplexification of this maximal torus. We then choose our collection of simple rootsto be Π := α1, α2, . . . , αn, where αj : T → C∗ is defined by the property

αj(R(θ1, θ2, . . . , θn)) = ei(θj−θj+1)

for j ∈ 1, . . . , n− 1, while αn : T → C∗ satisfies

αn(R(θ1, θ2, . . . , θn)) = ei(θn−1+θn).

The highest root λ is then given by

λ(R(θ1, θ2, . . . , θn)) = ei(θ1+θ2).

Furthermore, the subset of simple roots orthogonal to λ is Λ = Π \ α2.

Now, let B : C2n ⊗C2n → C be the complexification of the dot product on R2n. Onethen has the Grassmannian of isotropic 2-planes in C2n, GrB(2,C2n). More explicitly,

GrB(2,C2n) := V ∈ Gr(2,C2n) : V ⊆ V⊥,

where V⊥ denotes the complement of V ∈ Gr(2,C2n) with respect to B. One can verifythat GrB(2,C2n) has a point whose SO2n(C)-stabilizer is PΛ, so that

SO2n(C)/PΛ ∼= GrB(2,C2n).

By (6.3), we have another SO2n(C)-equivariant isomorphism

P(Omin) ∼= GrB(2,C2n), (6.10)

where Omin is the minimal nilpotent orbit of SO2n(C).

It remains to give the contact structure on GrB(2,C2n) for which (6.10) is an iso-morphism of contact varieties. In other words, it remains to find the unique SO2n(C)-invariant contact structure on GrB(2,C2n). To this end, let F denote the tautologicalbundle on GrB(2,C2n), whose fibre over V ∈ GrB(2,C2n) is V itself. Note that F is asubbundle of the trivial bundle GrB(2,C2n) × C2n, so that we may consider the sub-bundle F⊥ of complements with respect to B. By definition, F ⊆ F⊥, and we may


defineE := Hom(F,F⊥/F).

Note that E is canonically a subbundle of Hom(F,O⊕2n/F), the pullback toGrB(2,C2n)of TGr(2,C2n). In fact, we have the inclusion

E ⊆ TGrB(2,C2n)

of subbundles of Hom(F,O⊕2n/F), giving rise to a short exact sequence

0→ E → TGrB(2,C2n) → ∧2(F∨) → 0 (6.11)

(see [38], Chapter 14). Since ∧2(F∨) = det(F∨) is a line bundle, E is a corank-1 sub-bundle of TGrB(2,C2n). Indeed, we have the following proposition.

Proposition 69. The subbundle E ⊆ TGrB(2,C2n) is the unique SO2n(C)-invariant contactstructure on GrB(2,C2n).

Proof. By Proposition 62 and the discussion at the end of Section 6.2, the SO2n(C)-invariant contact structure on GrB(2,C2n) is the unique subbundle of H ⊆ TGrB(2,C2n)

such H is SO2n(C)-invariant and TGrB(2,C2n)/H is the ample generator of Pic(GrB(2,C2n)).Accordingly, it will suffice to prove that E possesses these two properties. For theformer, note that F⊥/F is an SO2n(C)-invariant subbundle of O⊕2n/F. Hence, E =

Hom(F,F⊥/F) is an SO2n(C)-invariant subbundle of Hom(F,O⊕2n/F), and thereforealso of TGrB(2,C2n). For our second property, (6.11) gives a bundle isomorphism

TGrB(2,C2n)/E∼= det(F∨).

The bundle det(F∨) is indeed the ample generator of Pic(GrB(2,C2n)), so our proof iscomplete.

We wish to conclude with a comparison of our presentation of the SO2n(C)-invariantcontact structure on the isotropic Grassmannian to the one presented in [44] (pp.353–354), whose distribution we will denote by P. There, GrB(2,C2n) is given an alter-native presentation, as a parameter space for lines in a hyperquadric of dimension2n − 2. If ` is line in the hyperquadric representing a point in the parameter space,and if we choose an isomorphism ` ∼= P1, the fibre P` is the space of global sections ofthe (2n − 4)-fold direct sum of the hyperplane bundle on the P1. One must choose anisomorphism for each point in order to describe P and so this description — while ex-


plicit — is local. Our presentation of the unique SO2n(C)-invariant contact structure,with distribution E given above, does not depend on a family of isomorphisms anduses the tautological bundle on the isotropic Grassmannian directly.

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University of Torontoblog.math.toronto.edu/GraduateBlog/files/2016/03/Thesis3.pdfAbstract The Equivariant Geometry of Nilpotent Orbits and Associated Varieties Peter Crooks Doctor