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

References in corpus (3)

• Cell decompositions of character varieties
• On Character Varieties with Non-Connected Structure Groups
• On the $E$-polynomial of parabolic ${\mathrm{Sp}}_{2n}$-character varieties