Mixed Hodge polynomials of character varieties

TL;DR

This paper computes the mixed Hodge polynomials of certain character varieties by counting points over finite fields, leading to geometric insights and conjectures about their cohomology, which are proven for the case n=2.

Contribution

It introduces a method to calculate E-polynomials of character varieties using finite field point counting and character tables, and formulates conjectures on their cohomology structure.

Findings

Calculated E-polynomials for specific character varieties.

Deduced topological Euler characteristics.

Proved conjectures for n=2.

Abstract

We calculate the E-polynomials of certain twisted GL(n,C)-character varieties M_n of Riemann surfaces by counting points over finite fields using the character table of the finite group of Lie-type GL(n,F_q) and a theorem proved in the appendix by N. Katz. We deduce from this calculation several geometric results, for example, the value of the topological Euler characteristic of the associated PGL(n,C)-character variety. The calculation also leads to several conjectures about the cohomology of M_n: an explicit conjecture for its mixed Hodge polynomial; a conjectured curious Hard Lefschetz theorem and a conjecture relating the pure part to absolutely indecomposable representations of a certain quiver. We prove these conjectures for n = 2.