E-polynomials of character varieties for real curves

Thomas John Baird, Michael Lennox Wong

Published 2020-06-01Version 1

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 (\ref{GenFunctBig}) 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 \cite{HRV}.