Hodge-Deligne polynomials of character varieties of free abelian groups

Let F be a finite group and X be a complex quasi-projective F-variety. For r is an element of N, we consider the mixed Hodge-Deligne polynomials of quotients X-r/F, where F acts diagonally, and compute them for certain classes of varieties X with simple mixed Hodge structures (MHSs). A particularly interesting case is when X is the maximal torus of an affine reductive group G, and F is its Weyl group. As an application, we obtain explicit formulas for the Hodge-Deligne and E-polynomials of (the distinguished component of) G-character varieties of free abelian groups. In the cases G = GL(n, C) and SL(n, C), we get even more concrete expressions for these polynomials, using the combinatorics of partitions.

Automated summary

The mixed Hodge-Deligne polynomials on complex quasi-projective F-varieties were studied and calculated for varieties with simple mixed Hodge structures. Particularly focused on the case of the maximal torus of an affine reductive group and its Weyl group, explicit formulas for G-character varieties of free abelian groups were obtained as an application, with concrete expressions derived for GL(n, C) and SL(n, C) using partition combinatorics.