Filtrations on Springer fiber cohomology and Kostka polynomials

Bellamy, G. and Schedler, T. (2018) Filtrations on Springer fiber cohomology and Kostka polynomials. Letters in Mathematical Physics, 108(3), pp. 679-698. (doi: 10.1007/s11005-017-1002-7)

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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.

Item Type:Articles
Glasgow Author(s) Enlighten ID:Bellamy, Professor Gwyn
Authors: Bellamy, G., and Schedler, T.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Letters in Mathematical Physics
Publisher:Springer Verlag
ISSN (Online):1573-0530
Published Online:26 September 2017
Copyright Holders:Copyright © 2017 The Authors
First Published:First published in Letters in Mathematical Physics 108(3):679-698
Publisher Policy:Reproduced under a Creative Commons License

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Project CodeAward NoProject NamePrincipal InvestigatorFunder's NameFunder RefLead Dept
662571Symplectic representation theoryGwyn BellamyEngineering and Physical Sciences Research Council (EPSRC)EP/N005058/1M&S - MATHEMATICS