IASM Preprint Series 2022-01: E-polynomials of generic GLn⋊<σ> -character varieties: branched case
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 GLn o<σ>- character varieties. We restrict the monodromies around the ramification points to generic semi-simple conjugacy classes contained in GLn σ, and compute the E-polynomials of these character varieties using the character table of GLn(q) o<σ>. The result is expressed as the inner product of certain symmetric functions associated to the wreath products (Z/2Z) N oSN. We are then led to a conjectural formula for the mixed Hodge polynomial, which involves (modified) Macdonald polynomials and wreath Macdonald polynomials.