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

摘要

We calculate the E-polynomials of certain twisted GL(n,ℂ)-character varieties M_nmathcal{M}_{n} of Riemann surfaces by counting points over finite fields using the character table of the finite group of Lie-type textGL(n,mathbbF_q)text{GL}(n,mathbb{F}_{q}) 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 M_nmathcal{M}_{n}: 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.