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 Broué-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.
From: Minh-Tâm Trinh [view email]
[v1]
Sat, 9 Aug 2025 17:28:37 UTC (18 KB)
[v2]
Fri, 11 Sep 2026 04:54:37 UTC (25 KB)
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