On the mixed Hodge polynomials of character varieties of Riemann surfaces
Seminar Room 1, Newton Institute
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.