Fernando Rodriguez-Villegas (UT Austin)

Title: "Mixed Hodge polynomials of character varieties"

Thanks to the Weil conjectures (proved by Deligne) we know that counting points of varieties over finite fields yields topological information about them. In this talk I will describe such a calculation for certain character varieties, parameterizing representations of the fundamental group of a Riemann surface into GL_n.

I will discuss the main ingredients of the calculation, which involves an array of techniques from combinatorics and representation theory of finite groups of Lie type. The outcome of the calculation has several geometric consequences about the varieties; for example, it allows us to compute their topological Euler characteristic. But more importantly it naturally leads to interesting conjectures about their mixed Hodge polynomials and a surprising relation to the cohomology of certain quiver varieties.

In the general case, where we prescribe conjugacy classes for the image of loops around a finite number of punctures on the surface, the geometry of the character varieties turns out to be related to the Macdonald polynomials of combinatorics.

This is joint work with T. Hausel and E. Letellier