Filtrations on Springer fiber cohomology and Kostka polynomials

Gwyn Bellamy,Travis Schedler

doi:10.1007/s11005-017-1002-7

Gwyn Bellamy, Travis Schedler

Open Access

https://doi.org/10.1007/s11005-017-1002-7

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Abstract

We prove a conjecture which expresses the bigraded Poisson-de Rham homology of the nilpotent cone of a semisimple Lie algebra in terms of the generalized (one-variable) Kostka polynomials, via a formula suggested by Lusztig. This allows us to construct a canonical family of filtrations on the flag variety cohomology, and hence on irreducible representations of the Weyl group, whose Hilbert series are given by the generalized Kostka polynomials. We deduce consequences for the cohomology of all Springer fibers. In particular, this computes the grading on the zeroth Poisson homology of all classical finite W-algebras, as well as the filtration on the zeroth Hochschild homology of all quantum finite W-algebras, and we generalize to all homology degrees. As a consequence, we deduce a conjecture of Proudfoot on symplectic duality, relating in type A the Poisson homology of Slodowy slices to the intersection cohomology of nilpotent orbit closures. In the last section, we give an analogue of our main theorem in the setting of mirabolic D-modules.

Highlights

Let g be a semisimple complex Lie algebra, N ⊆ g∗ the nilpotent cone, W the Weyl group, and G a connected complex Lie group with Lie G = g
The graded multiplicity space of each irreducible representation χ of W has Hilbert series given by the generalized Kostka polynomial Kg,χ (t), which in the case of g = sln is an ordinary one-variable Kostka polynomial
The above results allow us to deduce the grading on the zeroth Poisson homology, HP0(O(Sφ ∩ N )) := O(Sφ ∩ N )/{O(Sφ ∩ N ), O(Sφ ∩ N )} = HP0DR(Sφ ∩ N ), as well as the filtration on the quantizations of Sφ ∩ N, which are quantum W-algebras

Summary

Introduction

We go further and produce canonical filtrations on the cohomology of the flag variety whose Hilbert series is given in (1.1): Theorem 1.3 For every element λ ∈ h∗reg, there is a canonical associated filtration Fλ on H 2 dim B−∗(T ∗B) whose associated graded vector space is HP∗DR(N ) This is W -equivariant: Fw(λ) = w(Fλ). The above results allow us to deduce the grading on the zeroth Poisson homology, HP0(O(Sφ ∩ N )) := O(Sφ ∩ N )/{O(Sφ ∩ N ), O(Sφ ∩ N )} = HP0DR(Sφ ∩ N ), as well as the filtration on the quantizations of Sφ ∩ N , which are (centrally reduced) quantum W-algebras These naturally assign to the top cohomology of each Springer fiber ρ−1(φ) a h∗reg-family of filtrations whose Hilbert series we compute. We use this to generalize this result to the mirabolic setting, i.e., the setting of SLn-equivariant D-modules on sln × Cn

Springer fibers and W -algebras

Hochschild cohomology of quantum W-algebras

Higher cohomology of the Springer fiber

Proudfoot’s conjecture on symplectic duality

Recollections on D-modules on N

Hotta and Kashiwara’s theorem

Proofs of results on W -algebras and Slodowy slices

Hamiltonian reduction and a mirabolic generalization

The mirabolic case