E-polynomials of character varieties for real curves

Abstract

We calculate the E-polynomial for a class of the (complex) character varieties ${\mathcal{M}}_n^{\tau }$ associated to a genus $g$ Riemann surface $\Sigma$ equipped with an orientation reversing involution $\tau$. Our formula expresses the generating function ${\sum }_{n=1}^{\infty }E({\mathcal{M}}_n^{\tau })T^n$ as the plethystic logarithm of a product of sums indexed by Young diagrams. The proof uses point counting over finite fields, emulating Hausel and Rodriguez-Villegas.

39 pages, incorporated referee comments and divided the former section 7 into current sections 7 and 8

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