E-Polynomials of Generic GL_n\rtimes ~-Character Varieties: Unbranched Case

2022/03/02 by Shu, Cheng
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2203.00856

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 GL_n\rtimes ~-character varieties. We introduce k>0 punctures on the surface, and 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.

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