E-Polynomials of Generic $\mathbf {\operatorname {\mathrm {GL}}_n\rtimes \!<\!\sigma \!>\!}~$ -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 $\operatorname {\mathrm {GL}}_n\rtimes \!}$ -character varieties. We restrict the monodromies around the branch points to generic semi-simple conjugacy classes contained in $\operatorname {\mathrm {GL}}_n\sigma $ and compute the E-polynomials of these character varieties using the character table of $\operatorname {\mathrm {GL}}_n(q)\rtimes \!\!$ . The result is expressed as the inner product of certain symmetric functions associated to the wreath product $(\mathbb {Z}/2\mathbb {Z})^N\rtimes \mathfrak {S}_N$ . We are then led to a conjectural formula for the mixed Hodge polynomial, which involves (modified) Macdonald polynomials and wreath Macdonald polynomials.

Keywords

• 14M35
• 20C33
• 05E05