Hodge-Deligne polynomials of character varieties of free abelian groups

• Main Authors: Florentino Carlos, Silva Jaime
• Format: Article
• Language: English
• Published: De Gruyter 2021-05-01
• Series: Open Mathematics
• Subjects: free abelian group; character variety; mixed hodge structures; hodge-deligne polynomials; equivariant e-polynomials; finite quotients; 32s35; 20c30; 14l30
• Online Access: https://doi.org/10.1515/math-2021-0038

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