Traces on finite w-algebras

TRACES ON FINITE W-ALGEBRAS
PAVEL ETINGOF AND TRAVIS SCHEDLER Abstract. We compute the space of Poisson traces on a classical W-algebra modulo an arbitrary central character, i.e., linear functionals on such an algebra invariant under Hamiltonian derivations. This space identifies with the top cohomology of the corresponding Springer fiber. As a consequence, we deduce that the zeroth Hochschildhomology 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 theentire cohomology of the Springer fiber identifies with the so-called Poisson-de Rham homology (defined previously by the authors) of the classical W-algebra modulo a central character.

arXiv:1004.4634v1 [math.RT] 26 Apr 2010

1. Introduction The main goal of this note is to compute the zeroth Poisson homology of classical finite W-algebras modulo a central character, and the zeroth Hochschildhomology of their quantizations. 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), and earlier results of the authors on the characterization of zeroth Poisson homology in terms of D-modules. This implies anupper 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 using positive characteristic arguments. We also show that the Poisson-de Rham homology groups of classical W-algebras (defined earlier by the authors) are isomorphic to the cohomology groups of the Springer fiber in complementarydimension. 1.1. Definition of classical W-algebras. We first recall the definition of classical Walgebras (see, e.g., [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-Morozovtheorem, 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 ωe (x, y) := e, [x, y] . This restricts to a symplectic form on g−1 . Fix a Lagrangian l ⊂ g−1 , and set (1.1.1) me := l ⊕
i≤−2

gi .

Date: April 22, 2010.
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Then, wedefine a shift of me by e: (1.1.2) me := {x − e, x : x ∈ me } ⊂ Sym g. The classical W-algebra We is defined to be the Hamiltonian reduction of g with respect to me and the zero locus of me , i.e., (1.1.3) We := (Sym g/me · Sym g)me . 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 aPoisson algebra. The bracket { , } : We ⊗ 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 (Sym g)g , by the embedding (Sym g)g → (Sym g)me → (Sym g/(me Sym g))me . It is known that this composition is injective (since, by Kostant’s theorem, the coset e + me meets generic semisimple coadjoint orbits ofg). Let Z := (Sym g)g and Z+ = (g Sym g)g be its augmentation ideal. We therefore have an embedding Z → We , and can consider the quotient, called the finite W -algebra, (1.1.4)
0 We := We /Z+ We .

Since we will only work with finite W-algebras, we will omit the word “finite” from now on. 1.2. The Springer correspondence. We need to recall a version of the Springer correspondence between...