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Hodge-Deligne polynomials of character varieties of abelian groups

Carlos A. Florentino, Jaime A. M. Silva

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.

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.

Paper Structure

This paper contains 20 sections, 32 theorems, 91 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries on Character Varieties and on Mixed Hodge Structures
  3. Character varieties
  4. Mixed Hodge structures
  5. Hodge polynomials and balanced varieties
  6. Separably pure, elementary and round varieties
  7. Elementary and separably pure varieties
  8. Round varieties
  9. Cohomology and Mixed Hodge Structures for Finite Quotients
  10. Equivariant mixed Hodge structures
  11. Cohomology of finite quotients
  12. Character formulae
  13. Abelian character varieties and their Hodge-Deligne polynomials
  14. Mixed Hodge polynomials of abelian character varieties
  15. Normality of the distinguished component
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $r\geq1$, $G$ be a complex reductive group with maximal torus $T$ and Weyl group $W$. Then, where $A_{g}$ is the automorphism induced on $H^{1}(T,\mathbb{C})$ by $g\in W$, and $I$ is the identity automorphism.

Figures (1)

  • Figure 1: Venn diagram with several classes of MHS

Theorems & Definitions (83)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • Definition 2.2
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Definition 2.5
  • Example 2.6
  • ...and 73 more