Hodge-Deligne polynomials of character varieties of abelian groups

Abstract

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

Rephrased some results in section 5 (in particular, Lemma 5.5 was corrected, and Remark 5.6 added); other minor improvements

References in corpus (3)

• On the E-polynomials of a family of Character Varieties
• A lax monoidal Topological Quantum Field Theory for representation varieties
• Homological stability for spaces of commuting elements in Lie groups

Cited by in corpus (8)

• Flawed groups and the topology of character varieties
• Representation varieties of twisted Hopf links
• Plethystic exponential calculus and characteristic polynomials of permutations
• Virtual Classes of Representation Varieties of Upper Triangular Matrices via Topological Quantum Field Theories
• Mixed Hodge structures on character varieties of nilpotent groups
• $e$-polynomials of character varieties
• On the Motivic Class of the Moduli Stack of Twisted $G$-Covers
• Tensor Products of Quiver Bundles