Level-Rank Dualities for Finite Reductive Groups

TL;DR

This paper provides evidence for dualities in the representation theory of finite reductive groups, linking blocks of cyclotomic Hecke algebras through level-rank duality phenomena expressed via partition combinatorics.

Contribution

It introduces new dualities between blocks of cyclotomic Hecke algebras associated with finite reductive groups, extending the level-rank duality to this setting.

Findings

Identifies dualities between blocks of cyclotomic Hecke algebras for finite reductive groups.

Expresses these dualities explicitly in terms of partition combinatorics.

Recovers known level-rank duality phenomena in a new algebraic context.

Abstract

This is an extended abstract of our work "Level-Rank Dualities from $\Phi$-Cuspidal Pairs..." We present evidence for a family of surprising coincidences within the representation theory of a finite reductive group $G$: more precisely, dualities between blocks of cyclotomic Hecke algebras attached by Brou\'e-Malle to $\Phi$-cuspidal pairs of $G$, where the Hecke parameters are specialized not to the order of the underlying finite field, but to roots of unity. For the groups $G = \mathrm{GL}_n(\mathbf{F}_q)$, these coincidences can be expressed very concretely in terms of the combinatorics of partitions, and the whole story recovers an avatar of the level-rank duality studied by Frenkel, Uglov, Chuang-Miyachi, and others.