E-series of character varieties of non-orientable surfaces

TL;DR

This paper studies two types of character varieties associated with non-orientable surfaces, computing their point counts over finite fields and deriving formulas for their E-series, including special cases like the real projective plane.

Contribution

It introduces explicit computations of E-series for character varieties of non-orientable surfaces, extending understanding of their geometric and arithmetic properties.

Findings

Derived formulas for point counts over finite fields

Computed E-series for specific non-orientable surfaces

Analyzed the case of the real projective plane with one puncture

Abstract

In this paper we are interested in two kinds of (stacky) character varieties associated to a compact non-orientable surface. (A) We consider the quotient stack of the space of representations of the fundamental group of this surface to GL(n). (B) We choose a set of k-punctures on the surface and a generic k-tuple of semisimple conjugacy classes of GL(n), and we consider the stack of anti-invariant local systems on the orientation cover of the surface with local monodromies around the punctures given by the prescribed conjugacy classes. We compute the number of points of these spaces over finite fields from which we get a formula for their E-series (a certain specialization of the mixed Poincar\'e series). In case (B), we discuss the mixed Poincar\'e series when the surface is the real projective plane and k=1.