E-polynomials of generic GL n ⋊ 〈 σ 〉 -character varieties: unbranched case

For any unbranched double covering of a compact Riemann surface, we study the associated character varieties that are unitary in the global sense, which we call GL n ⋊ 〈 σ 〉 -character varieties. We introduce k > 0 punctures on the surface, restrict the monodromies around the punctures to generic semi-simple conjugacy classes in GL n , and compute the E-polynomials of these character varieties using the character table of 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.