### On the mixed Hodge polynomials of character varieties of Riemann surfaces

Rodriguez-Villegas, F *(Texas at Austin)*

Thursday 17 December 2009, 11:30-12:30

Seminar Room 1, Newton Institute

#### Abstract

I will describe a calculation of the number of points over finite fields of the varieties of the title (parameterizing generic representations of the fundamental group of a Riemann surface into GL_n). Since the varieties are polynomial count this calculation yields their E-polynomial (a geometric invariant).
The results are best expressed in terms of a generating function involving the Macdonald polynomials of combinatorics. We conjecture that a natural deformation of these formulas in fact gives the full mixed Hodge polynomial of the varieties.
This is joint work with T. Hausel and E. Letellier.

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