Transformation Groups, Vol. 15, No. 4, 2010, pp. 843-850 ©Birkhauser Boston (2010)

TRACES ON FINITE W-ALGEBRAS

PAVEL ETINGOF

TRAVIS SCHEDLER

Department of Mathematics

Department of Mathematics

Cambridge, MA 02139, USA etingof@math.mit.edu

Cambridge, MA 02139, USA trasched@gmail.com

Dedicated to Vladimir Morozov on the 100th anniversary of his birth

Abstract. We compute the space of Poisson traces on a classical W-algebra, i.e., linear functionals invariant under Hamiltonian derivations. Modulo any central character, this space identifies with the top cohomology of the corresponding Springer fiber. As a consequence, we deduce that the zeroth Hochschild homology of the corresponding quantum W-algebra modulo a central character identifies with the top cohomology of the corresponding Springer fiber. This implies that the number of irreducible finite-dimensional representations of this algebra is bounded by the dimension of this top cohomology, which was established earlier by C. Dodd using reduction to positive characteristic. Finally, we prove that the entire cohomology of the Springer fiber identifies with the so-called Poisson-de Rham homology (defined previously by the authors) of the centrally reduced classical W-algebra.

The main goal of this paper is to compute the zeroth Poisson homology of classical finite W-algebras, and the zeroth Hochschild homology of their quantizations. Modulo any central character, both spaces turn out to be isomorphic to the top cohomology of the corresponding Springer fiber. The proof is based on the presentation of the Springer D-module on the nilpotent cone by generators and relations (due to Hotta and Kashiwara [HK84]), and earlier results of the authors on the characterization of zeroth Poisson homology in terms of D-modules. This implies an upper bound on the number of irreducible finite-dimensional representations of a quantum W-algebra with a fixed central character, which was previously established by C. Dodd [Dod10] using positive characteristic arguments. We also show that the Poisson-de Rham homology groups of a centrally reduced classical W-algebra (defined earlier by the authors) are isomorphic to the cohomology groups of the Springer fiber in complementary dimension.

Acknowledgments. We are grateful to Roman Bezrukavnikov and Ivan Losev for useful discussions. The first author's work was partially supported by NSF grant DMS-0504847. The second author is a five-year fellow of the American Institute

1. Introduction

DOI: 10.1007/s00031-010-9103-8

Received April 23, 2010. Accepted May 17, 2010. Published online June 26,2010.

of Mathematics, and was partially supported by the ARRA-funded NSF grant DMS-0900233.

1.1. Definition of classical W-algebras

We first recall the definition of classical W-algebras (see, e.g., [GG02], [Los10] and the references therein). Let g be a finite-dimensional simple Lie algebra over C with the nondegenerate invariant form (—, —). We will identify g and g* using this form. Let G be the adjoint group corresponding to g. Fix a nilpotent element e € g. By the Jacobson-Morozov theorem, there exists an sl2-triple (e, h, f), i.e., elements of g satisfying [e, f ] = h, [h, e] = 2e, and [h, f ] = —2f. For i € Z, let gi denote the h-eigenspace of g of eigenvalue i. Equip g with the skew-symmetric form we(x,y) := (e, [x, y]). This restricts us to a symplectic form on g_i. Fix a Lagrangian l C g_i, and set

me := l 0 0 gi. (1.1)

Then we define a shift of me by e:

me := {x — (e, x) : x € me} C Symg. (1.2)

The classical W-algebra We is defined to be the Hamiltonian reduction of g with respect to me and the character (e, •), i.e.,

We := (Sym g/me • Sym g)me, (1.3)

where the invariants are taken with respect to the adjoint action. It is well-known that, up to isomorphism, We is independent of the choice of the sl2-triple containing e.

Since it is a Hamiltonian reduction, We is naturally a Poisson algebra. The bracket { , } : We <E> We ^ We is induced by the standard bracket on Sym g. The Poisson center of We (i.e., elements z such that {z, F} =0 for all F) is isomorphic to (Symg)g, by the embedding (Symg)g ^ (Symg)me ^ (Symg/(me Symg))me. It is known that this composition is injective (since, by Kostant's theorem, the coset e + me meets generic semisimple coadjoint orbits of g).

Let Z := (Symg)g and Z+ = (g Symg)g be its augmentation ideal. We therefore have an embedding Z ^ We, and can consider the central quotient

W0 := We/Z+We. (1.4)

1.2. The Springer correspondence

We need to recall a version of the Springer correspondence between representations of the Weyl group W of g and certain G-equivariant local systems on nilpotent orbits in g.

Let N C g be the nilpotent cone. Let B be the flag variety of g, consisting of Borel subalgebras b C g. Consider the Grothendieck-Springer map p : g := {(b,g) : g € b} C Bx g ^ g, which restricts to the Springer resolution N := p-1(N) ^ N. Note that N = T*B.

Let W be the Weyl group of g and Irrep( W) its set of irreducible representations, up to isomorphism. For each x € Irrep(W), denote by Vx the underlying vector space and by x : W ^ Aut(VX) the corresponding representation.

Then there is a well-known isomorphism (e.g., [Spr78, Theorem 1.13])

HdimR p-1(e)(p-1(e)) = 0 ^ ® Vx, (1.5)

X£lrrepe(W)

where Irrep(W) = [J/G Irrepe(W) and, for all x € Irrepe(W), is a certain irreducible representation of the component group n0(StabG(e)) of the stabilizer of e in G. For each x € Irrepe(W), let us use the notation Ox := O(e) = G • e.

1.3. The main results

For any Poisson algebra A, we consider the zeroth Poisson homology, HP0(A) := A/{A, A} (which is the same as the zeroth Lie homology). Its dual is the space of Poisson traces, i.e., linear functionals A ^ C which are invariant under Hamilto-nian derivations {a, -}.

As a consequence of [GG02] (see also [Los10, §2.6]), there is a natural action of the stabilizer StabG(e, h, f) of the sl2-triple (e, h, f) on We by Poisson automorphisms. This is because of the alternative construction of We in [GG02] which is invariant under StabG(e, h, f): We = (Sym g/n^ • Sym g)pe, where n'e = {x — (e, x) :

X € 0i<-2 gi} and Pe = 0i<_1 gi.

Since this action on We is Hamiltonian, it gives rise to an action of the component groups n0(StabG(e, h, f)) on HP0(We). Note that, since StabG(e, h, f) is the reductive part of StabG(e), the component group coincides with n0(StabG(e)). Clearly, this group also acts on HP0(W°).

Our first main result is the following theorem.

Theorem 1.6. As ni(StabG(e))-representations,

HP0(We°) = HdimR(p-1 (e))(p-1(e))= 0 ^x ® Vx- (1.7)

x£lrreps(W)

Here the action on the right-hand side is in the first component.

Remark 1.8. There is a slightly different way to view the Springer correspondence through [HK84] which further explains the above results. Namely, for a smooth variety X, denote by the right D-module of volume forms on X. Then [HK84, Theorem 5.3] states that

)= 0 Mx ® Vx, (1.9)

x£lrrep(W)

where Mx are irreducible, holonomic, pairwise nonisomorphic G-equivariant right D-modules on N.

Each D-module Mx is uniquely determined by its support, which is the closure, O, of a nilpotent coadjoint orbit O = O(e) C N (i.e., a symplectic leaf of N),

together with a G-equivariant local system on O (the restriction of Mx to O). Then, O = Ox, and the local system is .

Taking the pushforward of (1.9) to a point, one can deduce that HPo(W0) is isomorphic to the right-hand side of (1.7) using the method of [ES09] recalled in Section 2 below.

Next, let Ug denote the universal enveloping algebra of g, and let Wf := (Ug/meUg)me be the quantum W-algebra, which is a filtered (in general, non-commutative) algebra whose associated graded algebra is We, as a Poisson algebra. The center of Weq is an isomorphic image of Z(Ug), which is identified with Z as an algebra via the Harish-Chandra isomorphism. Let n : Z 4 C be a character, and define the algebras W^ := We/(ker(n)) and Wf'n := Wf/(ker(n)). These are filtered Poisson (resp., associative) algebras whose associated graded algebras are W^1. Moreover, using the construction of [GG02] as above (i.e., Wf = (Ug/n/Ug)pe), Wf as well as Wf'n admit actions of StabG(e, h, f) (as does Wn for all n). Since this action is Hamiltonian, HP0(Wn) and HH0(Wf'n) admit actions of n0 (StabG(e)) = n0(StabG(e, h, f)) for all n.

Consider the zeroth Hochschild homology HH0(Wf'n) := Wf'n/[Wf'n, Wf'n]. There is a canonical surjection HP0(W°) ^ grHH0(Wf'n).

Theorem 1.10.

(i) The canonical surjection HP0(W°) 4 grHH0(Wf'n) is an isomorphism.

(ii) The families HP0(Wn) and HH0(Wf'n) are flat in n. In particular, for all n, they are isomorphic to the top cohomology of the Springer fiber,

as representations of n0 (Stabc(e)).

(iii) The groups HP0(We) and HH0(Wf), considered as Z[n0(Stab^(e))]-modules, are

isomorphic to Z <g> HdimR p (e)(p-1(e)).

Theorem 1.10 follows from Theorem 1.13, as explained below.

Corollary 1.11 (C. Dodd, [Dod10]). For every central character n, the number of distinct irreducible finite-dimensional representations of Wf'n is at most dim HdimR p-1 (e) (p-1(e)).

Proof. This immediately follows from the above theorem, because the number of isomorphism classes of irreducible finite-dimensional representations of any associative algebra A is dominated by dim A/ [A, A] (since characters of nonisomorphic irreducible representations are linearly independent functionals on A/[A, A]). □

Remark 1.12. The argument in the appendix to [ES09] by I. Losev together with Corollary 1.11 also implies an upper bound on the number Ne of prime (or, equiv-alently, primitive) ideals in Wf'n. For every nilpotent orbit Oe whose closure contains e, let Me e/ denote the number of irreducible components of the intersection Oe n Spec We of the closure of the orbit Oe with the Kostant-Slodowy slice to e. Then,

Ne < J2 Me,e' • dimHdimRp-1 (e')(p_1(e')),

where the sum is over the distinct orbits Oe/ whose closure contains e. Briefly, we explain this as follows: Losev's appendix to [ES09] gives a map from finite-dimensional irreducible representations of W|,'n to prime ideals of W|'n supported on the irreducible component of O, n Spec We containing e', and shows that all prime ideals are constructed in this way. (More precisely, in op. cit., a construction is given of all prime ideals of filtered quantizations of affine Poisson varieties with finitely many symplectic leaves, which specializes to this one since the aforementioned irreducible components coincide with the symplectic leaves of Spec W,1, and W|'n and W|,'n are quantizations of W,0 and W^, respectively.) Then, the bound follows from Corollary 1.11.

1.4. Higher homology

Finally, following [ES09], one may consider the higher Poisson-de Rham homology groups, HPfR (Wn), of Wn, whose definition we recall in the following section. Here, we only need to know that HPDR(A) = HP0(A) for all Poisson algebras A (although the same is not true for higher homology groups). Let n : Z ^ C be an arbitrary central character.

The following theorem is a direct generalization of Theorem 1.6. Hence, we only prove this theorem, and omit the proof of the aforementioned theorem.

Theorem 1.13. As no(StabG(e))-representations,

HPfR(Wen) = HdimR p-1 (e)-i(p-1(e)).

Moreover, for generic n, HHi(W|'n) is also isomorphic to these.

Remark 1.14. For e = 0 one has W| = Ug, and the algebras W|'n are the maximal primitive quotients of Ug. In this case, the last statement of Theorem 1.13 holds for all regular characters n (see [Soe96], [VdB98], and [VdB02]). On the other hand, the genericity assumption for n cannot be removed for i > 0. Namely, for nonregular values of n, it is, in general, not true that HHi(W|'n) is isomorphic to the cohomology HdimR p 1 (e)-i(p-1(e)) of the Springer fiber. For example, when e = 0 in g = sl2, then the variety Spec W,0 is the cone C2/Z2. In this case, by [FSSA03, Theorem 2.1] (and the preceding comments), HHi(W|'n) = 0 for all i > 3 when n : Z (Ug) ^ C is the special central character corresponding to the Verma module with highest weight —1, i.e., the character corresponding to the fixed point of the Cartan h under the shifted Weyl group action.

2. The construction of [ES09]

We prove Theorem 1.13 using the method of [ES09], which we now recall.

To a smooth affine Poisson variety X, we attached the right D-module MX on X defined as the quotient of the algebra of differential operators DX by the right ideal generated by Hamiltonian vector fields. Then HP0(OX) identifies with the (underived) pushforward MX ®dx Ox of MX to a point.

More generally, if X is not necessarily smooth, but equipped with a closed embedding i : X ^ V into a smooth affine variety V (which need not be Poisson), we defined the right D-module MX i on V as the quotient of DV by the right ideal

generated by functions on V vanishing on X and vector fields on V tangent to X which restrict on X to Hamiltonian vector fields. This is independent of the choice of embedding, in the sense that the resulting D-modules on V supported on X correspond to the same D-module on X (up to a canonical isomorphism) via Kashiwara's theorem. Call this D-module MX. The pushforward of MX to a point remains isomorphic to HP0(OX).

More generally, for an arbitrary affine variety X, we defined the groups HPDR(CX) as the full (left derived) pushforward of MX to a point.

3. Proof of Theorem 1.13

Our main tool is

Theorem 3.1 ([HK84, Theorem 4.2 and Prop. 4.8.1(2)]; see also [LS97, §7]).

Mn = )• (3.2)

We now begin the proof of Theorem 1.13. First, take n = 0. Since Mn = ), it follows that, letting n and n denote the projections of N and N to a

point,

HPfR(On) = L^*(Mn) = L^n) - Hdimc^(N).

Similarly, if we consider Spec W° C N (the intersection of a Kostant-Slodowy slice to the orbit of e with N), then it follows, viewing all varieties as embedded in the smooth variety g, that p*(Qp—1(specW0}) = Mspecwo, since p_1(Spec W°) C N is smooth. In more detail, let Ye denote a formal neighborhood of Spec W° in N, let V denote a formal neighborhood of p_1(Spec W°) in N, and let [e, g] denote a formal completion of [e, g] at 0. Then Ye — p_1(Spec W°) x [e, g] and Ye = Spec W° x [e, g]. With these identifications, p|y- = p|p-i(Spec wo} x Id-— Then = ^p— i(Specw°} ^ ] and M7e - ^Specw0 ^ ^q. Since Mn = ),

restricting to V yields MYe = ), and we conclude from the above that

MSpec W° = 1 (Spec W0}).

Since p_1(Spec W0) 4 Spec W0 is birational, we have dimC p_1(Spec W0) = dimC Spec W0. Hence, HPfR(W0) = Hdimc Spec wo0_ (p_1(Spec We0)). Next, observe that the contracting C*-action on Spec W° lifts to a deformation retraction of p_1(Spec W°) to p_1(e), as topological spaces (in the complex topology). Moreover, p_1(e) is compact, and hence its dimension must equal the degree of the top cohomology, dimR p_1(e) = dimC p_1(Spec W°) = dimC Spec W° (This can also be computed directly: all of these quantities are equal to the complex codimension of G • e inside N.) We conclude the first equality of the theorem for n = 0, i.e.,

HP?R(W0) = H dimR p—1(e}(p_1(e)).

Since the parameter space of n has a contracting C* action with fixed point n = 0, to prove flatness of HPfR(Wn), it suffices to show that dimHPfR(Wn) = dim HPDR(W°) for generic n. For generic n, Spec W^ is smooth and symplectic,

and hence (by [ES09, Example 2.2]), MSpecwn = QSpecwn, so that HPfR(Wn) =

HdimSpec we_i(Spec Wen). Moreover, p_1(Spec W^) 4 Spec W^. Next, for all n,

the family p 1(Spec W,n) is topologically trivial [Slo80a] (see also [Slo80b]), and hence its cohomology has constant dimension, and equals

dim Hdimc Spec W0-i (p-1(Spec We0)).

Hence, for generic n, dim HPDR(W,n) = dim HPD5R(W,0), as desired.

Let us now prove the second statement of the theorem. Let ft be a formal parameter. For any central character n : Z ^ C, consider the character n/ft : Z((ft)) ^ C((ft)). Let W|'n/R := W1 ((ft))/ker(n/ft). As we will explain below, by the results of Nest and Tsygan [NT95] and Brylinski [Bry88], when Spec W,n is smooth (which is true for generic n), HHi(W|'n/R) = Hdimc Wn-i(Spec Wn, C((ft))). Hence, HHi(W|'n/R) = HPfR(We0)((ft)) for generic n. This implies the statement.1

In more detail, W,f'n/R is obtained from a deformation quantization of W,n in the following way. Let Wes be the ft-adically completed Rees algebra (J)m>0ftmFmW|, where F*W| is the filtration on W|. This is a deformation quantization of We. Consider the quotient Wg'n := Wg/(ker(n)). Then Wg'n is a deformation quantization of W,n. (Recall that, in general, a deformation quantization of a Poisson algebra A0 is an algebra of the form An = (A0[[ft]],*), A0[[ft]] := {^ i>0 aifti, ai € A0} satisfying a * b = ab + O(ft) and a * b — b * a = ft{a, b} + O(ft2), up to an isomorphism.) Then, by [Bry88, Theorems 2.2.1 and 3.1.1] and [NT95, Theorem A2.1], if Wn is smooth and symplectic, then HH^W^[ft-1]) = Hdimc Wen-i(Spec Wn, C((ft))). Furthermore, the map defined by x ^ ftx, x € g defines an isomorphism Wg'n[ft-1] ^ W|,n/^. Since Wn is smooth for generic n, this implies the results claimed in the previous paragraph. □

4. Proof of Theorem 1.10

Note that (iii) easily follows from (ii), since HP0(We) is a finitely generated graded Z-module, which is flat (i.e., projective) by Theorem 1.13, and Z is non-negatively graded with Z0 = C.

Thus, it suffices to prove that dim HH0(W|-n) = dimHdimRp-1(e)(p-1(e)) for all central characters n. As remarked before the statement of Theorem 1.10, there is a canonical surjection HP0(W0) ^ grHH0(W|'n). Hence, for all n, dim HH0(W|,n) < dim HP0(W0), which equals dimHdimR p-1(e)(p-1(e)) by Theorem 1.6 (which follows from Theorem 1.13). The minimum value of dim HH0(W|'n) is attained for generic n (since HH0(W|) is a finitely generated Z-module), where it is also dimHdimRp (e)(p-1(e)) by Theorem 1.13. Hence, this dimension must be the same for all n. □

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