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July 2020 Poincaré polynomials of character varieties, Macdonald polynomials and affine Springer fibers
Anton Mellit
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Ann. of Math. (2) 192(1): 165-228 (July 2020). DOI: 10.4007/annals.2020.192.1.3

Abstract

We prove an explicit formula for the Poincaré polynomials of parabolic character varieties of Riemann surfaces with semisimple local monodromies, which was conjectured by Hausel, Letellier and Rodriguez-Villegas. Using an approach of Mozgovoy and Schiffmann the problem is reduced to counting pairs of a parabolic vector bundle and a nilpotent endomorphism of prescribed generic type. The generating function counting these pairs is shown to be a product of Macdonald polynomials and the function counting pairs without parabolic structure. The modified Macdonald polynomial $\tilde H_\lambda [X;q,t]$ is interpreted as a weighted count of points of the affine Springer fiber over the constant nilpotent matrix of type $\lambda$.

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Anton Mellit. "Poincaré polynomials of character varieties, Macdonald polynomials and affine Springer fibers." Ann. of Math. (2) 192 (1) 165 - 228, July 2020. https://doi.org/10.4007/annals.2020.192.1.3

Information

Published: July 2020
First available in Project Euclid: 21 December 2021

Digital Object Identifier: 10.4007/annals.2020.192.1.3

Subjects:
Primary: 14H60

Keywords: Character varieties , Hall algebras , Higgs bundles , Macdonald polynomials , Poincaré polynomials

Rights: Copyright © 2020 Department of Mathematics, Princeton University

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Vol.192 • No. 1 • July 2020
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