  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.
