Level-Rank Dualities for Finite Reductive Groups

Abstract

This is an extended abstract of our work "Level-Rank Dualities from $\hat{I}¦$-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 Broué-Malle to $\hat{I}¦$-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.

14 pages