Multiplicity of characters of finite reductive groups and Drinfeld doubles

TL;DR

This paper investigates the multiplicities of tensor products of characters in finite reductive groups, explores Frobenius Schur indicators of Drinfeld double modules, and raises questions about character tensor product detection methods.

Contribution

It provides explicit computations of tensor product multiplicities and analyzes Frobenius Schur indicators, advancing understanding of character ring structures and representation theory of finite reductive groups.

Findings

Computed multiplicities of tensor products of characters.

Analyzed Frobenius Schur indicators for Drinfeld double modules.

Raised questions on detecting tensor product non-vanishing via Weyl group characters.

Abstract

In this paper, we compute the multiplicities of tensor products of almost unipotent characters and Deligne Lusztig characters of a finite reductive group $G^F$, and these multiplicities are related to the ring structure of the complex irreducible characters of $G^F$. In addition, we consider Frobenius Schur indicators of modules over the Drinfeld doubles of finite reductive groups. In the final section, we study the multiplicities of tensor products of almost unipotent characters and pose the question of whether their non vanishing can be detected through the multiplicities of tensor products of irreducible characters of the Weyl group.