Thomas Gerber (Aachen), Harish-Chandra series for finite unitary groups and crystal graphs

Abstract:

Harish-Chandra theory is a tool to label the simple modules of a finite group of Lie type $G$, in non-defining characteristic. In this talk, we focus on the case where $G$ is a finite unitary group of index $n$, and we aim at determining the Harish-Chandra series. In this context, it is known that it is sufficient to understand the unipotent modules, which are labelled by partitions of $n$. This gives rise to a combinatorial approach.

After introducing the Harish-Chandra branching graph for unipotent $G$-modules on the one hand, and the crystal graph of the level 2 Fock space on the other hand, we explain how these two graphs are conjecturally related, and what new information this brings. This is joint work with Gerhard Hiss and Nicolas Jacon.