Hodge-Deligne polynomials of character varieties of free abelian groups

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.

Generating series for the E‐polynomials of GL(n,C)$GL(n,{\mathbb {C}})$‐character varieties

“…6.1] (and [5, Remarks 6.2]) and also appears in [18, Prop. 1.9]; a more detailed proof has been recently presented in [10], so we refer to those proofs, adding only a couple of comments that may serve to deduce the present statement. The weight polynomial used by Dimca‐Lehrer is equivalent to the E ‐polynomial in the case of Hodge–Tate type varieties, using the substitution $$t^{2}=uv$$.…”

“…For example, using $$B_{1}^{\Gamma }(x)=(x-1)^{r}$$ in Corollary 4.11, we recover the E ‐polynomials of the $$GL_{n}$$‐character varieties of $$\Gamma =\mathbb {Z}^{r}$$, the free abelian group of rank r . See [10] and Subsection 5.1 below. (3)We thank Mozgovoy for drawing our attention to his recent Preprint [21], where a similar formula to (4.7) is shown (cf., [21, Thm. 1.2]), within a general framework for counting isomorphism classes of objects in additive categories over finite fields (see also [20] and [13, Appendix]).…”

“…We start by examining representations of finitely presented abelian groups. The following result, recently obtained in [10], deals with the group $$\Gamma =\mathbb {Z}^{r}$$. Theorem Let $$r,n\in \mathbb {N}$$.…”

“…Proof For the Hodge–Tate statement, see [10] or Proposition 5.3, below. In that article, the formula is shown as a consequence of computing the mixed Hodge–Deligne polynomial of $$\mathcal {X}_{\mathbb {Z}^{r}}GL_{n}$$.…”

“…As a first application of these results, in Section 5, we write the E ‐polynomial of the abelian stratum $$\mathcal {X}_{\Gamma }^{[1^{n}]}GL_{n}\subset \mathcal {X}_{\Gamma }GL_{n}$$ in terms of usual partitions, generalizing a result in [10]; we also apply the same methods to write the E ‐polynomial of the so‐called Cartan brane on the moduli space of rank n and degree zero Higgs bundles, an algebraic variety that is generally not of polynomial type.…”