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.

中文翻译：

---

自由阿贝尔群的字符变体的Hodge-Deligne多项式

令FF为有限群，XX为复拟射影FF-变种。对于{\ mathbb {N}}中的r∈N r \，我们考虑商X r / F {X} ^ {r} \ hspace {-0.15em} \ text {/} \ hspace的混合Hodge-Deligne多项式{-0.08em} F，其中FF对角起作用，并针对具有简单混合Hodge结构（MHS）的某些类别的品种XX计算它们。一个特别有趣的情况是，当XX是仿射还原基团GG的最大圆环，而FF是它的Weyl基团时。作为一种应用，我们获得了自由阿贝尔族的GG-字符变体（的独特组成部分）的Hodge-Deligne和EE-多项式的显式公式。在G = GL（n，C）的情况下G = GL \ left（n，{\ mathbb {C}} \ hspace {-0.1em}）和SL（n，C）SL \ left（n，{\ mathbb {C}} \ hspace {-0.1em}）中，我们得到了这些多项式的更具体的表达式，