Arithmetic harmonic analysis on character and quiver varieties

Abstract

We propose a general conjecture for the mixed Hodge polynomial of the generic character varieties of representations of the fundamental group of a Riemann surface of genus $g$ to ${\mathrm{GL}}_n\left(\mathbb{C}\right)$ with fixed generic semisimple conjugacy classes at $k$ punctures. This conjecture generalizes the Cauchy identity for Macdonald polynomials and is a common generalization of two formulas that we prove in this paper. The first is a formula for the E-polynomial of these character varieties which we obtain using the character table of ${\mathrm{GL}}_n\left({\mathbb{F}}_q\right)$. We use this formula to compute the Euler characteristic of character varieties. The second formula gives the Poincaré polynomial of certain associated quiver varieties which we obtain using the character table of $\mathfrak{g}{\mathfrak{l}}_n\left({\mathbb{F}}_q\right)$. In the last main result we prove that the Poincaré polynomials of the quiver varieties equal certain multiplicities in the tensor product of irreducible characters of ${\mathrm{GL}}_n\left({\mathbb{F}}_q\right)$. As a consequence we find a curious connection between Kac-Moody algebras associated with comet-shaped, and typically wild, quivers and the representation theory of ${\mathrm{GL}}_n\left({\mathbb{F}}_q\right)$.