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