Arithmetic harmonic analysis on character and quiver varieties
Abstract
We present a conjecture generalizing the Cauchy formula for Macdonald polynomials. This conjecture encodes the mixed Hodge polynomials of the character varieties of representations of the fundamental group of a Riemann surface of genus g to GLn(C) with fixed generic semi-simple conjugacy classes at k punctures. Using the character table of GLn(Fq) we calculate the E-polynomial of these character varieties and confirm that it is as predicted by our main conjecture. Then, using the character table of gln(Fq), we calculate the E-polynomial of certain associated comet-shaped quiver varieties, the additive analogues of our character variety, and find that it is the pure part of our conjectured mixed Hodge polynomial. Finally, we observe that the pure part of our conjectured mixed Hodge polynomial also equals certain multiplicities in the tensor product of irreducible representations of GLn(Fq). This implies a curious connection between the representation theory of GLn(Fq) and Kac-Moody algebras associated with comet-shaped, typically wild, quivers.