E-polynomials of character varieties for real curves

Abstract

We calculate the E-polynomial for a class of (complex) character varieties M n τ \mathcal {M}_n^{\tau } associated to a genus g g Riemann surface Σ \Sigma equipped with an orientation reversing involution τ \tau . Our formula expresses the generating function ∑ n = 1 ∞ E ( M n τ ) T n \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 [Invent. Math. 174 (2008), pp. 555–624].