Multiplicity of characters of finite reductive groups and Drinfeld doubles
GyeongHyeon Nam
TL;DR
The paper develops a comprehensive framework for computing multiplicities of tensor products among Deligne–Lusztig and almost unipotent characters of finite reductive groups G^F, linking these multiplicities to the ring structure of Ĥ(G^F) and to invariants arising in Drinfeld doubles. It introduces a detailed type-based decomposition using pseudo-Levi subgroups, Green functions, Möbius inversion, and Deligne–Lusztig data to produce explicit formulas for single and multiple Deligne–Lusztig factors, including special cases such as the Steinberg character and split tori. It also analyzes Frobenius–Schur indicators for modules over the Drinfeld double and derives asymptotic degree and leading-term information for tensor squares, with connections to semisimple elements and Weyl-group data. The work highlights vanishing and non-vanishing criteria, provides polynomial-in-q counts in split cases, and poses a natural question about whether nonvanishing in the Weyl-group side implies nonvanishing in G^F for general reductive groups, extending known GL_n phenomena. Overall, the results advance the understanding of the representation theory of finite reductive groups, their Deligne–Lusztig characters, and the interplay with Drinfeld doubles and Weyl-group combinatorics.
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.