Poincare polynomials of character varieties, Macdonald polynomials and affine Springer fibers

TL;DR

This paper derives an explicit formula for the Poincaré polynomials of parabolic character varieties of Riemann surfaces, linking them to Macdonald polynomials and affine Springer fibers, thus advancing understanding of their geometric and combinatorial structures.

Contribution

It provides a new explicit formula for these polynomials, connecting character varieties, Macdonald polynomials, and affine Springer fibers, and confirms a conjecture by Hausel, Letellier, and Rodriguez-Villegas.

Findings

Explicit formula for Poincaré polynomials of character varieties

Connection between Macdonald polynomials and affine Springer fibers

Confirmation of conjecture by Hausel, Letellier, and Rodriguez-Villegas

Abstract

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 $\lambda$.