Filtrations on Springer fiber cohomology and Kostka polynomials

TL;DR

This paper establishes a deep connection between the cohomology of Springer fibers, Kostka polynomials, and W-algebras, providing new filtrations and confirming conjectures related to symplectic duality.

Contribution

It proves a conjecture linking Poisson-de Rham homology with Kostka polynomials and constructs canonical filtrations on cohomology and representations, extending to W-algebras and mirabolic D-modules.

Findings

Computed the grading on zeroth Poisson homology of classical finite W-algebras.

Established filtrations on Hochschild homology of quantum finite W-algebras.

Confirmed Proudfoot's conjecture on symplectic duality in type A.

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.