Poincaré polynomials of character varieties, Macdonald polynomials and affine Springer fibers
By Anton Mellit. Annals of Mathematics, 2017
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We prove an explicit formula for the Poincar\'e 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 bundles 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 $\lambda$.
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/427a5463b9f853094f2726e444a2cfaeaa10b351
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Citations in the corpus
- Poincaré polynomials of moduli spaces of Higgs bundles and character varieties (no punctures) (2017, methodology): “In the next paper [13], we extend Schiffmann’s [17] and Schiffmann– Mozgovoy’s [16] methods to the parabolic case.”