Filtrations on Springer fiber cohomology and Kostka polynomials
File(s)
Published version
Author(s)
Bellamy, G
Schedler, TJ
Type
Journal Article
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.
Date Issued
2018-03-01
Date Acceptance
2017-08-11
Citation
Letters in Mathematical Physics, 2018, 108 (3), pp.679-698
ISSN
0377-9017
Publisher
Springer Verlag
Start Page
679
End Page
698
Journal / Book Title
Letters in Mathematical Physics
Volume
108
Issue
3
Copyright Statement
© The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
License URL
Sponsor
National Science Foundation
Grant Number
DMS-1406553
Subjects
01 Mathematical Sciences
02 Physical Sciences
Mathematical Physics
Publication Status
Published
Date Publish Online
2017-09-26