  Abstract. Let G be a complex reductive group and A be an Abelian variety of
dimension d over $\mathbb{C}$. We determine the Poincar\'e polynomials and also
the mixed Hodge polynomials of the moduli space $\mathcal{M}_{A}^{H}(G)$ of
G-Higgs bundles over A. We show that these are normal varieties with symplectic
singularities, when G is a classical semisimple group. For
$G=GL_{n}(\mathbb{C})$, we also compute Poincar\'e polynomials of natural
desingularizations of $\mathcal{M}_{A}^{H}(G)$ and of G-character varieties of
free abelian groups, in some cases. In particular, explicit formulas are
obtained when dim A=d=1, and also for rank 2 and 3 Higgs bundles, for arbitrary
d>1.
