Abstract
Let
F
F
be a finite group and
X
X
be a complex quasi-projective
F
F
-variety. For
r
∈
N
r\in {\mathbb{N}}
, we consider the mixed Hodge-Deligne polynomials of quotients
X
r
/
F
{X}^{r}\hspace{-0.15em}\text{/}\hspace{-0.08em}F
, where
F
F
acts diagonally, and compute them for certain classes of varieties
X
X
with simple mixed Hodge structures (MHSs). A particularly interesting case is when
X
X
is the maximal torus of an affine reductive group
G
G
, and
F
F
is its Weyl group. As an application, we obtain explicit formulas for the Hodge-Deligne and
E
E
-polynomials of (the distinguished component of)
G
G
-character varieties of free abelian groups. In the cases
G
=
G
L
(
n
,
C
)
G=GL\left(n,{\mathbb{C}}\hspace{-0.1em})
and
S
L
(
n
,
C
)
SL\left(n,{\mathbb{C}}\hspace{-0.1em})
, we get even more concrete expressions for these polynomials, using the combinatorics of partitions.