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

TL;DR

This paper computes the E-polynomials of certain unitary character varieties associated with branched double covers of Riemann surfaces, using character tables and symmetric functions, and proposes a conjectural formula for their mixed Hodge polynomials.

Contribution

It introduces a method to compute E-polynomials of $ ext{GL}_n times<\sigma>$-character varieties in the branched case, connecting them to symmetric functions and conjecturing their mixed Hodge polynomials.

Findings

E-polynomials expressed via symmetric functions and character tables

Conjectural formula for mixed Hodge polynomial involving Macdonald polynomials

Results extend understanding of character varieties in branched covering scenarios

Abstract

For any branched double covering of compact Riemann surfaces, we consider the associated character varieties that are unitary in the global sense, which we call $\text{GL}_n\rtimes\!<\!\sigma\!>\!~$-character varieties. We restrict the monodromies around the ramification points to generic semi-simple conjugacy classes contained in $\text{GL}_n\sigma$, and compute the E-polynomials of these character varieties using the character table of $\text{GL}_n(q)\rtimes\!<\!\sigma\!>\!$. The result is expressed as the inner product of certain symmetric functions associated to the wreath products $(\mathbb{Z}/2\mathbb{Z})^N\rtimes\mathfrak{S}_N$. We are then led to a conjectural formula for the mixed Hodge polynomial, which involves (modified) Macdonald polynomials and wreath Macdonald polynomials.