Motivic Mirror Symmetry for Character Stacks

TL;DR

This paper develops a motivic version of topological mirror symmetry for character stacks of semisimple groups, generalizing previous results and exploring dualities and automorphisms, with a specific example for SL(2).

Contribution

It introduces a motivic framework for mirror symmetry in character stacks, extending cell decomposition and duality concepts to arbitrary reductive groups.

Findings

Generalized Mellit's cell decomposition to reductive groups

Described automorphisms on character stacks

Established duality via Weil pairing for cells

Abstract

We propose a motivic version of T. Hausel and M. Thaddeus' Topological Mirror Symmetry for character stacks associated with arbitrary semisimple groups, which is an analogue of F. Loeser and D. Wyss' result for Chow motives of moduli spaces of Higgs bundles. As first steps towards it, we generalize A. Mellit's cell decomposition to arbitrary connected and reductive groups. We use it to describe all automorphisms on the associated character stacks. Then we show that the Weil pairing induces a duality between cells that interchanges automorphisms by connected components. As a toy example, we show that these results imply our conjecture for the special linear group of rank two.