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Counting stringy points on a family of character varieties

Lucas de Amorin, Martin Mereb

Abstract

We provided explicit formulas for the number of stringy points over finite fields of parabolic type A character varieties with generic semisimple monodromy. This leads to formulas for their stringy E-polynomials. In particular, they satisfy the Betti Topological Mirror Symmetry Conjecture of T. Hausel and M. Thaddeus, as well as a refinement regarding isotypic components. Our proof is based on a Frobenius' type formula for Clifford's type settings and an analysis of it in a specific set-up related to regular wreath products with cyclic groups.

Counting stringy points on a family of character varieties

Abstract

We provided explicit formulas for the number of stringy points over finite fields of parabolic type A character varieties with generic semisimple monodromy. This leads to formulas for their stringy E-polynomials. In particular, they satisfy the Betti Topological Mirror Symmetry Conjecture of T. Hausel and M. Thaddeus, as well as a refinement regarding isotypic components. Our proof is based on a Frobenius' type formula for Clifford's type settings and an analysis of it in a specific set-up related to regular wreath products with cyclic groups.

Paper Structure

This paper contains 25 sections, 47 theorems, 241 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Clifford's theory
  4. Characters of finite general linear groups
  5. E-polynomials
  6. Katz's and Groechenig--Wyss--Ziegler's theorems
  7. Type A parabolic character varieties
  8. A character formula
  9. The formula
  10. Proof of theorem \ref{['character-formula']}
  11. Proof of proposition \ref{['magic-formula']}
  12. The case of block-like groups
  13. Set up
  14. Classification of fixed characters
  15. Ramification index
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $n$ be a natural number. For any divisor $d$ of $n$, $F_d$-discrete torsion $\mathcal{D}$ of order $\frac{d}{K}$, and any generic semisimple conjugacy class $\mathcal{C}$ of $\operatorname{SL}_n(\overline{\mathbb{Q}})$, where $\tau$ runs over all multi-partitions types of size $n$, $\Phi_g$ is an arithmetic function, $\mathcal{H}_{\tau'}$ are polynomials associated to $\tau$, and $C_{s,\tau}$

Theorems & Definitions (86)

  • Theorem 1.1
  • Corollary 1.1
  • Theorem 1.2
  • Corollary 1.2
  • Corollary 1.3
  • Theorem 1.3
  • Theorem 2.1: Theorem 1 of Green2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 76 more