Hodge Deligne polynomials of abelian character varieties
Let F be a finite group and X be a complex quasi projective F variety. For r in 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. 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 formulae 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.