E-Polynomials of Generic $\text{GL}_n\rtimes\!<\!\sigma\!>\!~$-Character Varieties: Unbranched Case

TL;DR

This paper computes E-polynomials for a class of unbranched double cover character varieties of Riemann surfaces, introducing new formulas involving symmetric functions and conjecturing a mixed Hodge polynomial expression.

Contribution

It introduces a novel approach to calculating E-polynomials of $ ext{GL}_n times<\sigma>$-character varieties using symmetric functions and proposes a conjectural formula for their mixed Hodge polynomials.

Findings

E-polynomials expressed via symmetric function inner products

Derived formulas for unbranched double cover character varieties

Proposed conjectural formula for mixed Hodge polynomial

Abstract

For any unbranched double covering of compact Riemann surfaces, we study the associated character varieties that are unitary in the global sense, which we call $\text{GL}_n\rtimes\!<\!\sigma\!>\!~$-character varieties. We introduce $k>0$ punctures on the surface, and restrict the monodromies around the punctures to generic semi-simple conjugacy classes in $\text{GL}_n$, and compute the E-polynomials of these character varieties using the character table of $\text{GL}_n(q)$. The result is expressed as the inner product of certain symmetric functions. We are then led to a conjectural formula for the mixed Hodge polynomial, which is built out of (modified) Macdonald polynomials, their self-pairings, and self-pairings of wreath Macdonald polynomials.