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
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