E-Polynomials of Generic $\text{GL}_n\rtimes\!<\!\sigma\!>\!~$-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 $\text{GL}_n\rtimes\!<\!\sigma\!>\!~$-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.