Mixed Hodge polynomials of character varieties: With an appendix by Nicholas M. Katz

We calculate the E -polynomials of certain twisted GL( n ,ℂ)-character varieties of Riemann surfaces by counting points over finite fields using the character table of the finite group of Lie-type and a theorem proved in the appendix by N. Katz. We deduce from this calculation several geometric results, for example, the value of the topological Euler characteristic of the associated PGL( n ,ℂ)-character variety. The calculation also leads to several conjectures about the cohomology of : an explicit conjecture for its mixed Hodge polynomial; a conjectured curious hard Lefschetz theorem and a conjecture relating the pure part to absolutely indecomposable representations of a certain quiver. We prove these conjectures for n =2.