Representation Homology

Yuri Berest

Cornell University

July 18, 2016

Plan

1. Motivation

2. Representation functor

3. Machinery: homotopical algebra

4. Representation homology of Lie algebras

5. Derived commuting schemes and Macdonald conjectures

6. Representation homology of spaces∗

1

References

1. Y.B., G.Khachatryan, A.Ramadoss, Derived representation schemesand cyclic homology, Adv. Math. 245 (2013), 625–689.

2. Y.B., A.Ramadoss, Stable representation homology and Koszul duality,J. Reine Angew. Math. 715 (2016), 143–187.

3. Y.B., G.Felder, A.Patotski, A.Ramadoss, T.Willwacher, Representationhomology, Lie algebra cohomology and the derived Harish-Chandrahomomorphism, JAMS (2016), to appear; arXiv:1410.0 043.

4. Y.B., G.Felder, A.Patotski, A.Ramadoss, T.Willwacher, Chern-Simonsforms and higher characters of Lie representations, IMRN (2016), toappear; arXiv:1505.05377.

5. Y.B., G.Felder, A.Ramadoss, Derived representation schemes andnoncommutative geometry, Contem. Math. 607 (2014), 113–162.

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1. Motivation

Throughout k denotes a field of characteristic zero.

Noncommutative Geometry

There is a natural way to associate to any commutative algebra A ageometric object: an affine k-scheme Spec(A).

Spec : Comm Algk → Schemesk

Attempts to extend this functor to the category of all associative algebrashave been largely unsuccessful.

To get around this problem M. Kontsevich and A. Rosenberg (2000)proposed a heuristic idea which turned out to be surprisingly useful.

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For an associative k-algebra A and a fixed integer n ≥ 0, let

Repn(A) = space of all n-dimensional k-linear representations of A

Repn(A) is naturally an affine k-scheme called the n-th representationscheme of A.

The representation scheme carries a natural GLn-action: its equivariantgeometry is closely related to the representation theory of A.

If k is algebraically closed and A is finitely generated, the orbits in Repn(A)are in bijection with the isomorphism classes of representations; the closedorbits correspond to the semisimple representations and parametrized bythe character scheme

Repn(A)//GLn := Spec k[Repn(A)]GLn .

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Now, for each n ≥ 0, we have the contravariant functor

Repn : Algk → Schemesk

Kontsevich-Rosenberg ‘Principle’: A property or noncommutativegeometric structure on an algebra A should induce a correspondinggeometric property or structure on all representation spaces Repn(A).

Thus, the family of functors Repnn≥0 may be viewed as a kind ofapproximation to ‘Spec’ on the category of associative algebras.

Applications: NC smooth spaces (Cuntz-Quillen), formal structuresand NC thickenings (Kapranov), NC symplectic structures (Kontsevich),NC bysimplectic geometry (Crawley-Boevey-Etingof-Ginzburg), doublePoisson brackets and NC quasi-Hamiltonian spaces (Van den Bergh),...

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Example (Cuntz-Quillen). An associative algebra A is called (formally)

smooth if Ω1(A) := Ker[A ⊗ A m→ A] is a projective bimodule. If A is asmooth, then Repn(A) is a (formally) smooth scheme for all n ≥ 0.

However, there are many interesting properties of noncommutative algebrasfor which the KR principle does not work. For example, such properties ofA as being NC complete intersection, Cohen-Macaulay, Calabi-Yau, ... donot induce the corresponding geometric properties of Repn(A).

One reason for this seems to be that the representation functor Repn isnot ‘exact’: it should be replaced by a (non-abelian) derived functor DRepn(much in the same way as ‘Hom’ should be replaced by ‘RHom’ or ‘⊗’ by⊗L in classical homological algebra.)

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We will see that DRepn(A) is represented by a commutative differentialgraded (DG) algebra k[DRepn(A)], which is well defined up to homotopy.

The homology of k[DRepn(A)] depends only on A and n, with

H0(k[DRepn(A)]) ∼= k[Repn(A)] .

We call H∗[DRepn(A)] the n-dimensional representation homology of A.

The higher representation homology obstructs Repn(A) from having thedesired property of A and thus measures the failure of the Kontsevich-Rosenberg ‘approximation’.

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For example, if A is a formally smooth algebra, we prove in [BFR] that

Hi[DRepn(A)] = 0 , ∀ i > 0

This explains why Repn preserves formal smoothness in any dimension n.

In contrast, if A is a noncommutative complete intersection (in the senseof Anick, Golod-Shafarevich, . . . ), then Repn(A) may or may not be acomplete intersection in the usual geometric sense, depending on whetherHi[DRepn(A)] vanish or not for i > 0 in a given dimension n.

(For a precise statement and further examples, see [BFR].)

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Derived characters of finite-dimensional representations

Representation homology allows one to extend the classical character theoryin a natural way. The key point is a relation to cyclic homology (see [BKR]).

Recall that the characters of finite-dimensional representations of an algebraA can be viewed as functions on its representation spaces: for each n ≥ 0,we have the following natural map

Trn(A) : A/[A,A]→ k[Repn(A)] , a 7→ [% 7→ Tr %(a)]

By linearity, this map extends to an algebra homomorphism

Sym Trn(A) : Symk(A/[A,A])→ k[Repn(A)] .

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Theorem (Procesi). The image of Sym Trn(A) is k[Repn(A)]GLn.

Observe that A/[A,A] = HC0(A) is the 0-th cyclic homology of A.

It turns out that there exist natural linear maps (morphisms of functors)

Trn(A)p : HCp(A)→ Hp[DRepn(A)] , ∀ p ≥ 0 ,

extending Trn(A) to higher cyclic homology. It is natural to think of (thevalues of) these maps as derived characters of representations.

The maps Trn(A)i assemble to a map of graded commutative algebras

ΛTrn(A)• : Λk [HC•(A)]→ H•[DRepn(A)]GLn .

In view of Procesi’s Theorem, it is natural to ask

Question: Is the map ΛTrn(A)• surjective?

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To address the question we first stabilize the family of maps ΛTrn(A)•passing to the limit n→∞.

Stabilization Theorem (see [BR]) says that, for an augmented algebraA, the stable trace map ΛTr∞(A)• is injective and (essentially) surjective:more precisely, the image of ΛTr∞(A)• is a DG subalgebra DRep∞(A)GL∞

0 ,which is dense in an appropriate inverse limit topology on DRep∞(A)GL∞.

Consequence: The (reduced) cyclic homology of A can be recovered fromits derived representation schemes:

Λ[HC•(A)] ∼= H•[DRep∞(A)GL∞0 ]

For a fixed finite n, comparing DRepn(A)GLn to DRep∞(A)GL∞0 , we find

homological obstructions to surjectivity: thus, in general, the answer tothe above question is negative. Procesi’s Theorem does not extend to thederived setting (see [BR]).

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Analogy: The (reduced version of) Loday-Quillen-Tsygan isomorphism

H•(gl∞(A), gl∞(k), k) ∼= Λ (HC•(A)[1])

Stabilization phenomenon

Hn(gln(A), gln(k), k) ∼= Hn(gln+1(A), gln+1(k), k) ∼= . . . ∼= Hn(gl∞(A), gl∞(k), k)

CompareΛ[HC•(A)] ∼= H•[DRep∞(A)GL∞

0 ]

Theorem (see [BR]). The DG algebra DRep∞(A)GL∞0 is Koszul dual to

the Chevalley-Eilenberg DG coalgebra C•(gl∞(A), gl∞(k), k).

In particular, this gives an intrinsic characterization of DRep∞(A)GL∞0 , up

to quasi-isomorphism.

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2. Representation functor

Representation homology can be defined for various types of algebras andgroups (e.g., for algebras over an arbitrary cyclic operad (see [BFPRW]),but perhaps most interestingly, it can be defined for arbitrary topologicalspaces. In this talk, we will just focus on (DG) Lie algebras and spaces.

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

Let g be a finite-dimensional Lie algebra over k. By a representation of an(arbitrary) Lie algebra a in g we mean a Lie algebra homomorphism a→ g .

The representation scheme Repg(a) is defined by its functor of points

Repg(a) : Comm Algk → Sets , B 7→ HomLie(a, g(B)) ,

assigning to a commutative k-algebra B the set of families of representationsof a in g parametrized by k-scheme Spec(B).

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The functor Repg(a) is represented by a commutative algebra ag, which hasthe following canonical presentation (see [3])

ag =Symk(a⊗ g∗)

〈〈 (x⊗ ξ(1)) · (y ⊗ ξ(2))− (y ⊗ ξ(1)) · (x⊗ ξ(2))− [x, y]⊗ x 〉〉.

Here g∗ is the vector space dual to g and ξ 7→ ξ(1) ∧ ξ(2) is the linear mapg∗ → ∧2g∗ dual to the Lie bracket on g.

Note that ag has a natural augmentation ε : ag → k induced by the zeromap a ⊗ g∗ → 0 . Thus a 7→ ag defines a functor with values in thecategory of augmented commutative algebras:

( – )g : Lie Algk → Comm Algk/k .

We call this functor the representation functor in g.

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Geometrically, one can think of the augmented algebra (ag, ε) as a coordinatering k[Repg(a)] of the based affine scheme Repg(a), with the basepointcorresponding to the trivial representation.

The Lie algebra g acts naturally on ag by derivations: this action is inducedby the coadjoint action ad∗g on g∗ and is functorial in a.

Write ( – )adgg : Lie Algk → Comm Algk/k for the subfunctor of ( – )g defined

by taking ad∗g-invariants:

aadgg := x ∈ ag : ad∗ξ(x) = 0 , ∀ ξ ∈ g .

If g is a reductive Lie algebra, aadgg represents the affine quotient schemeRepg(a)//G , where G is the associated algebraic group.

Our next goal is to explain how to derive the representation functor and itsinvariant subfunctor.

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3. Abstract homotopy theory

A model category is a category endowed with a certain structure that allowsone to do homological algebra in a non-abelian setting (Quillen).

Definition. A (closed) model category is a category C with threedistinguished classes of morphisms:

(i) weak equivalences or acyclic maps (∼→),

(ii) fibrations (),

(iii) cofibrations (→),

each of which is closed under composition and contains all identity maps.

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Axioms

MC1 C has all finite limits and colimits.

MC2 In any diagram Xg→ Y

f→ Z, if any two of the three maps f,g, and fg are weak equivalences, then so is the third.

MC3 Each of the three distinguished classes of maps is closed undertaking retracts: f is a retract of g if there is a commutativediagram

X //

f

X ′ //

g

X

f

Y // Y ′ // Y

such that the composition of the top and bottom rows is theidentity.

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MC4 In a commutative square

A -X

B?

∩

-....

........

........

.-

Y

??

if either of the two vertical maps is a weak equivalence, there isa lifting B → X making the diagram commute.

MC5 Any map A→ X in C may be factored in two ways:

(i) A∼→ B X ,

(ii) A → Y∼ X .

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Definition. A ∈ Ob(C) is called fibrant if A → ∗ is a fibration, andA ∈ Ob(C) is cofibrant if e→ A is a cofibration.

Assumption. We will assume that C is fibrant (i.e., all objects are fibrant).

A cylinder over A is an object Cyl(A) ∈ Ob(C) that factors (by MC5) thefolding map:

(id, id) : AqA → Cyl(A)∼→ A .

Two morphisms f, g : A→ X in C are (left) homotopic if

A⊂i1- Cyl(A)

i2 ⊃A

X

∃H?

..........g

f-

The map H is called a homotopy from f to g.

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Consequences of axioms

1. For any A ∈ Ob(C), there is a cofibrant object QA with an acyclic

fibration QA∼ A. This is called a cofibrant resolution of A.

2. For any two cofibrant resolutions QA, Q′A of A, there exist morphisms

QAf

gQ′A such that fg ∼ Id and gf ∼ Id.

3. Any morphism f : A→ B in C lifts to cofibrant resolutions, i. e.

QA

o

f// QB

o

Af

// B

The lifting f is unique up to homotopy.

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Homotopy category

The (‘classical’) homotopy category Ho(C) has the same objects as C, with

HomHo(C)(X,Y ) := HomC(QX, QY )/ ∼

where QX and QY are cofibrant replacements of X and Y .

There is a canonical quotient functor

γ : C→ Ho(C) , f 7→ [f ] .

Theorem. If F : C→ D sends weak equivalences to isomorphisms, thereis a unique F : Ho(C)→ D such that F γ = F.

Thus, Ho(C) = C[W−1] depends only on C and the class W = W (C) ofweak equivalences in C.

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Derived functors

Let F : C → D be a functor between two model categories. The (total)left derived functor LF of F is a right Kan extension of the functor

γD F : C→ D→ Ho(D)

along γC : C→ Ho(C) . Thus, by defintiion, LF is given by a pair (LF, t):

LF : Ho(C)→ Ho(D), t : LF γC → γD F ,

which is universal (in the obvious sense) among all such pairs.

Intuitively, LF provides the ‘best’ homotopical approximation to F ‘fromthe left’. LF may or may not exist, but when it exists, LF is determined(by F ) uniquely up to unique isomorphism. There is a dual notion of a rightderived functor RF obtained by reversing arrows in the above definition.

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Lemma/Definition. Let C and D be model categories. Let

F : C D : G

be a pair of adjoint functors. The following conditions are equivalent:

(a) F preserves cofibrations as well as acyclic cofibrations,

(b) G preserves fibrations as well as acyclic fibrations,

(c) F preserves cofibrations and G preserves fibrations.

A pair (F, G) satisfying these conditions is called a Quillen pair.

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Theorem (Quillen). Let F : C D : G be a Quillen pair. Then thederived functors LF and RG exist and form an adjoint pair

LF : Ho(C) Ho(D) : RG

In particular, LF is given by

LF (A) = γ F (QA) , LF (f) = γ F (f)

where QA∼ A is a cofibrant resolution in C and f : QA → QB is a

lifting of f : A→ B.

Lemma (Brown). If F : C → D carries acyclic cofibrations betweencofibrant objects in C to weak equivalences in D, then F preserves allweak equivalences between cofibrant objects. In that case, LF exists.

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Examples

DGA. Let DGAk be the category of associative DG algebras over a field kwith differential of degree (−1). DGAk has a model structure such that

i) the weak equivalences are the quasi-isomorphisms,

ii) the fibrations are the (degreewise) surjective maps,

iii) the cofibrations are the morphisms having the left lifting property w.r.t.acyclic fibrations.

iv) k is the initial object, and 0 is the terminal one. Thus DGAk is fibrant.

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DG Lie algebras. If char k = 0, the category DGLAk of DG Lie algebrasover k has a model structure with (i), (ii) and (iii) as above.

Commutative DGA. If char k = 0, the category CDGAk of commutativeDG algebras over k has a model structure with (i), (ii) and (iii) as above.

Non-negatively graded DGA. The categories DGA+k , CDGA+k and DGLA+khave model structures where the weak equivalences are the quasi-isomorphisms, the fibrations are the maps surjective in all positive degrees,and the cofibrations are retracts of almost free DG algebras.

Remark. If char k 6= 0, one should work with simplicial algebras instead ofdifferential graded algebras.

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4. Representation homology of Lie algebras

Recall that the representation functor is defined as a left adjoint functor:

( – )g : Lie Algk Comm Algk/k : g(–)

This adjunction extends naturally to the categories of DG algebras

( – )g : DGLAk CDGAk/k : g(–) (1)

Theorem. The functors (1) form a Quillen pair.

This follows from Quillen’s Lemma: indeed, g(–) is defined by tensoringg(B) = g ⊗ B over a field k, and this operation obviously preservesboth epimorphisms (fibrations) and surjective quasi-isomorphisms (acyclicfibrations).

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Thus, by Quillen’s Theorem, ( – )g has the total left derived functor

L( – )g : Ho(DGLAk)→ Ho(CDGAk/k)

which we call the derived representation functor in g.

For any a ∈ DGLAk, we now define

DRepg(a) := L(a)g

A more subtle argument (based Brown’s Lemma) shows the existence of

L( – )adgg : Ho(DGLAk)→ Ho(CDGAk/k) ,

Thus, we defineDRepg(a)ad g := L(a)ad g

g

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As in classical homological algebra, these derived functors are computed by

DRepg(a) ∼= (Qa)g , DRepg(a)adg ∼= (Qa)adg ,

where Qa is a(ny) cofibrant resolution of a in DGLAk.

Definition. The representation homology of a in g is defined by

H•(a, g) := H•[DRepg(a)]

This is a graded commutative k-algebra that depends only on a and g (noton the choice of resolution of a). If a ∈ Lie Algk is an ordinary Lie algebra,there is a natural algebra isomorphism

H0(a, g) ∼= ag

This justifies the name ‘derived representation scheme’ for DRepg(a).

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If g is a reductive Lie algebra, then we have

H•[DRepg(a)adg] ∼= H•(a, g)adg.

Thus, the homology of DRepg(a)adg can be viewed as the invariant part ofthe representation homology of a.

The derived Harish Chandra homomorphism

Assume that g is a finite-dimensional reductive Lie algebra. Let h be aCartan subalgebra of g, and W the corresponding Weyl group.

For any a, Reph(a) is naturally a closed subscheme of Repg(a). In fact, theinclusion h → g induces a morphism of schemes

Reph(a)/W → Repg(a)//G

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where Reph(a)/W is the quotient modulo the action of W on h.

This yields a morphism of functors ( – )adgg → ( – )Wh which extends to a

(unique) morphism of the derived functors: L( – )adgg → L( – )Wh . As aresult, for any DG Lie algebra a, we get a canonical map of commutativeDG algebras

Φg(a) : DRepg(a)adg → DReph(a)W . (2)

which we called the derived Harish-Chandra homomorphism.

Example. Let a be a one-dimensional Lie algebra. Since a is free, it iscofibrant in DGLAk. Hence DRepg(a) ∼= k[g] , DReph(a) ∼= k[h] , and Φg(a)is just the restriction map

k[g]G∼→ k[h]W .

This map is known to be an isomorphism (Chevalley Theorem).

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5. Derived commuting schemes

Let a be an abelian Lie algebra of dimension r ≥ 1. In this case,

Φg : DRepg(a)G → Λ[h∗ ⊗ (a⊕ ∧2a⊕ . . .⊕ ∧ra)]W ,

where ∧ia has homological degree i− 1.

If dim a = 2, Repg(a) is the classical commuting scheme of g, and thusDRepg(a) may be thought of the derived commuting scheme.

If dim a = 2, then Φg induces on the 0-th homology the map

k[Repg(a)]G → Sym[h∗ ⊗ a]W ,

which is known to be an isomorphism, at least when g is complex semisimpleand Repg(a)//G is reduced (Haiman).

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Conjecture 1. If a is abelian and dim(a) = 2, then, for any finite-dimensional reductive Lie algebra g, the derived Harish-Chandra map

Φg(a) : DRepg(a)G → Λ(h∗ ⊕ h∗ ⊕ h∗[1])W

is a quasi-isomorphism (at least when Repg(a)//G is reduced).

Remarks. 1. Conjecture holds for gl2 and sl2 as well as for gln, sln, sonand sp2n in the inductive limit as n→∞ (see [BFPRW])

2. If dimk(a) ≥ 3 , Φg(a) cannot be a quasi-isomorphism (for all g).

3. In the case of gln, Conjecture 1 says

DRepn(k[x, y])GL ∼= k[x1, . . . , xn, y1, . . . , yn, θ1, . . . , θn]Sn,

where the polynomial ring on the right is graded homologically so thatx1, . . . , xn and y1, . . . , yn have degree 0 , θ1, . . . , θn have degree 1 .

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Conjecture 1 can be restated in elementary terms. Consider the gradedcommutative algebra k[g×g]⊗∧g∗ , where k[g×g] is the ring of polynomialfunctions on g× g assigned homological degree 0, and ∧g∗ is the exterioralgebra of the dual Lie algebra g∗ assigned homological degree 1. Thedifferential on k[g× g]⊗ ∧g∗ is defined by

dϕ(ξ, η) := ϕ([ξ, η]) , ∀ (ξ, η) ∈ g× g , ∀ϕ ∈ g∗.

The DG algebra (k[g× g]⊗∧g∗, d) represents DRepg(a) in the homotopycategory of commutative DG algebras, and the derived Harish-Chandrahomomorphism Φg(a) is given in this case by the restriction map

(k[g× g]⊗ ∧g∗)G → (k[h× h]⊗ ∧h∗)W (3)

Conjecture 1 is thus equivalent to the claim that (3) is a quasi-isomorphism.

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The Constant Term Identity

Equating the bigraded Euler characteristics of both sides of Conjecture 1yields the following remarkable identity

(1− qt)l

(1− q)l(1− t)lCT

∏α∈R

(1− qteα)(1− eα)

(1− qeα)(1− teα)

=∑w∈W

det(1− qtw)

det(1− q w) det(1− t w)

Here R is a system of roots of g, l := dimk(h) is its rank, and CT :Z[Q]→ Z is the constant term map defined on the group ring of the rootlattice of R and extended to Z[Q][[q, t]] by linearity. The determinants onthe right are taken in the natural (reflection) representation of W on h.

Theorem (see [BFPRW]). The identity holds for gln and sln for all n .

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In the case of gln, the above constant term identity can be written indifferent combinatorial forms.

1. Let CT: Z[u±11 , . . . , u±1n ][[q, t]] → Z[[q, t]] be the constant term map.Then

1

n!

(1− qt)n

(1− q)n(1− t)nCT

∏1≤i6=j≤n

(1− qtui/uj)(1− ui/uj)(1− qui/uj)(1− tui/uj)

=

n∑j=0

tj

(q; q)n−j(t; t)j

where we use the standard q-calculus notation:

(v; q)∞ :=

∞∏j=0

(1− qjv) , (v; q)n :=(v; q)∞

(qnv; q)∞.

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2. Let Zn(q, t) be the left-hand side of the above identity and Z(v, q, t) =1 +

∑∞n=1Zn(q, t)vn the generating function. Then

Z(v, q, t) = exp

(∑λ

v|λ|

|λ|Wλ(q, t)

).

The sum is over all partitions λ = (λ1 ≥ λ2 ≥ · · · ≥ 0) of size |λ| =∑λi.

The coefficient Wλ(q, t) is a regularized product over the boxes (i, j) in theYoung diagram of the partition λ, with conjugate partition λ′ :

Wλ(q, t) =∏

(i,j)∈Y (λ)

′ (1− qjti)(1− q−j+1t−i+1)

(1− qλi−j+1t−λ′j+i)(1− q−λi+jtλ

′j−i+1

)

(In this product, we omit the factor 1− q0t0 appearing at (i, j) = (1, 1).)

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Then our identity takes the following simple form:

∑|λ|=n

Wλ(q, t) =1− qntn

(1− qn)(1− tn)(4)

Remark. Sums similar to the RHS of identity (4) occur in supersymmetricgauge theory: the Nekrasov instanton partition function of N = 2 superYang–Mills theory on R4 with U(N) gauge group.

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Macdonald Conjecture

In general, if k = C, the identity can be written in the integral form:

∫G

det(1 − qtAd g)

det(1 − qAd g) det(1 − tAd g)dg =

1

|W |∑w∈W

det(1− qtw)

det(1− qw) det(1− tw).

Here the integration is taken over a real compact form of the complex Liegroup G, which is equipped with the invariant Haar measure dg normalizedso that

∫Gdg = 1.

Notice that specializing t = 0 yields the well-known identity

∫G

dg

det(1 − qAd g)=

l∏i=1

1

1− qdi, (5)

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which exhibits the equality of the Poincare series of both sides of theChevalley isomorphism k[g]G ∼= k[h]W .

The Chevalley isomorphism has a natural ‘odd’ analogue: the Hopf-Koszul-Samelson isomorphism

(∧g)G ∼= ∧(Prim g) .

This classical isomorphism identifies the space of invariants in the exterioralgebra of g with the exterior algebra of its subspace of primitive elements.

At the level of Poincare series, the HKS isomorphism gives∫G

det(1 + qAd g) dg =

l∏i=1

(1 + q2di−1) , (6)

which may be viewed as an ‘odd’ analogue of (5).

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In his original 1982 paper on Macdonald conjectures, I. Macdonald observedthat (6) arises as a specialization of his famous identity

1

|W |CT

∏n≥0

∏α∈R

1− qneα

1− qnteα

=∏n≥0

l∏i=1

(1− qnt)(1− qn+1tdi−1)

(1− qn+1)(1− qntdi)(7)

In a remark after stating his conjecture (7), he asked whether (5) admits a(q, t)-generalization similar to (7).

It seems that our identity is an answer to Macdonald’s question.

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Question: Does the Macdonald identity (7) have a homological origin?

Hanlon and Fegin gave an interpretation in terms of cohomology of certainnilpotent Lie algebras: they made a precise conjecture on the structureof this cohomology that entails (7). The Hanlon-Feigin conjecture (a.k.athe strong Macdonald conjecture) was proved by Fishel, Grojnowski andTeleman (2008).

In [BFPRW], we give a different interpretation of the Macdonald identityin terms of representation homology that clarifies its relation to our HCconjecture.

We begin with a general remark. Working with DRepg(a), it is natural toput on a a homological grading. If a is abelian and dimk(a) = 2, gradinga amounts to splitting it into the sum of two one-dimensional subspaces ofhomological degrees p and r (we will write a = ap,r in this case).

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It turns out that the structure of DRepg(ap,r) essentially depends only onthe parities of p and r, and therefore there are 3 possibilities, which we referto as the even, mixed and odd cases (depending on whether p and r areboth even, have opposite parities or both odd).

In the even case, we show that the HC Conjecture holds for ap,r if and onlyif it holds for a with trivial grading (i.e., p = r = 0); thus, in this case, weexpect the Harish-Chandra homomorphism (2) to be a quasi-isomorphism.

In the mixed case, the situation is quite different: the derived Harish-Chandra homomorphism is no longer a quasi-isomorphism, and we need toconstruct a new map.

We will explain the construction for an arbitrary Lie algebra a.

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V. Drinfeld introduced a natural functor on the category of Lie algebrasthat associates to a Lie algebra a the universal invariant bilinear form:

λ(a) = Sym2k(a)/〈[x, y] · z − x · [y, z] : x, y, z ∈ a〉 .

Following a suggestion of Kontsevich, Getzler and Kapranov defined (ananalogue of) cyclic homology for Lie algebras as the non-abelian derivedfunctor of the Drinfeld functor λ.

Our staring point is a natural extension of the Drinfeld-Getzler-Kapranovconstruction: for an integer d ≥ 1, we consider the functor

λ(d) : DGLAk → Comk , a 7→ Symd(a)/[a,Symd(a)] ,

that assigns to a Lie algebra a (the target of) the universal invariantmultilinear form on a of degree d.

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Using Brown’s Lemma, we prove that, for any d , this functor has a leftderived functor defined on the homotopy category of DG Lie algebras:

Lλ(d) : Ho(DGLAk)→ Ho(Comk)

Definition. For d ≥ 1 , we set HC(d)• (Lie, a) := H•[Lλ

(d)(a)] .

Note that HC(1)• (Lie, a) ∼= H•+1(a, k) is the classical (Chevalley-Eilenberg)

homology of a, and HC(2)• (Lie, a) is the Getzler-Kapranov cyclic homology.

The meaning of our construction is clarified by the following

Theorem. The (reduced) cyclic homology of the universal envelopingalgebra U(a) of any DG Lie algebra a has a canonical Hodge-typedecomposition

HC•(Ua) =⊕d≥1

HC(d)• (Lie, a) .

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Next, for any homogeneous invariant polynomial on g of degree d, weconstruct natural trace maps

Tr(d)g (a) : Lλ(d)(a)→ DRepg(a)G ,

that are analogues of the derived character maps for Lie algebras.

Letting d run over the set d1. . . . , dl of fundamental degrees of g, we thendefine the homomorphism of commutative DG algebras

Λk[⊕li=1Tr(di)g (a)] : Λk[⊕li=1Lλ(di)(a)]→ DRepg(a)G,

which we call the Drinfeld trace map. In the simplest case when a is aone-dimensional Lie algebra, the Drinfeld trace map coincides the inverse ofthe Chevalley isomorphism.

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Theorem. If a has a mixed grading, the Drinfeld trace map is a quasi-isomorphism.

At the level of Euler characteristics, this quasi-isomorphism gives preciselythe classical Macdonald identity (7).

Unfortunately, our proof of the above theorem is not entirely self-contained:it relies on one of the main theorems of [FGT].

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For what kind of DG Lie algebras a the derived Harish Chandrahomomorphism (resp., the Drinfeld trace map) is a quasi-isomorphism?

A natural answer seems be given in topological terms: via Quillen’s RHTTheorem describing rational homotopy types of simply connected spaces interms of DG Lie algebra models. (This is joint work in progress with AjayRamadoss and Yeung Wai Kit.)

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