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

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 ${\text{GL}}_n⋊<\sigma >$-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.

arXiv admin note: text overlap with arXiv:2202.06506