Level-Rank Dualities from $Φ$-Cuspidal Pairs and Affine Springer Fibers
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| Format: | Preprint |
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2023
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| _version_ | 1866909645341196288 |
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| author | Trinh, Minh-Tâm Quang Xue, Ting |
| author_facet | Trinh, Minh-Tâm Quang Xue, Ting |
| contents | We propose a generalization of the level-rank dualities arising from Uglov's work on higher-level Fock spaces. The statements use Hecke algebras defined by Broué-Malle, which conjecturally describe the endomorphisms of Lusztig induction modules, and a generalization of Harish-Chandra theory due to Broué-Malle-Michel. For any generic finite reductive group $\mathbb{G}$ and integers $e, m > 0$, we conjecture that: (1) the intersection of a $Φ_e$-Harish-Chandra series and a $Φ_m$-Harish-Chandra series is parametrized by a union of blocks of the Hecke algebra of the $Φ_e$-cuspidal pair at an $m$th root of unity, and similarly for the Hecke algebra of the $Φ_m$-cuspidal pair at an $e$th root of unity; (2) these parametrizations match the blocks on the two sides; (3) when two blocks match, the bijection between them lifts to a derived equivalence between associated blocks of rational DAHAs. Surprisingly, these structures also appear in bimodules formed from the cohomology of affine Springer fibers studied by Oblomkov-Yun. When $\mathbb{G} = \mathbb{GL}_n$ and $e, m$ are coprime, we show that (1)-(3) hold, and that (3) recovers the level-rank dualities conjectured by Chuang-Miyachi and later proved through the work of several other people. Finally, we verify for many cases where $\mathbb{G}$ is exceptional that Broué-Malle's parameters are numerically compatible with our conjectures. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_17106 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Level-Rank Dualities from $Φ$-Cuspidal Pairs and Affine Springer Fibers Trinh, Minh-Tâm Quang Xue, Ting Representation Theory Algebraic Geometry We propose a generalization of the level-rank dualities arising from Uglov's work on higher-level Fock spaces. The statements use Hecke algebras defined by Broué-Malle, which conjecturally describe the endomorphisms of Lusztig induction modules, and a generalization of Harish-Chandra theory due to Broué-Malle-Michel. For any generic finite reductive group $\mathbb{G}$ and integers $e, m > 0$, we conjecture that: (1) the intersection of a $Φ_e$-Harish-Chandra series and a $Φ_m$-Harish-Chandra series is parametrized by a union of blocks of the Hecke algebra of the $Φ_e$-cuspidal pair at an $m$th root of unity, and similarly for the Hecke algebra of the $Φ_m$-cuspidal pair at an $e$th root of unity; (2) these parametrizations match the blocks on the two sides; (3) when two blocks match, the bijection between them lifts to a derived equivalence between associated blocks of rational DAHAs. Surprisingly, these structures also appear in bimodules formed from the cohomology of affine Springer fibers studied by Oblomkov-Yun. When $\mathbb{G} = \mathbb{GL}_n$ and $e, m$ are coprime, we show that (1)-(3) hold, and that (3) recovers the level-rank dualities conjectured by Chuang-Miyachi and later proved through the work of several other people. Finally, we verify for many cases where $\mathbb{G}$ is exceptional that Broué-Malle's parameters are numerically compatible with our conjectures. |
| title | Level-Rank Dualities from $Φ$-Cuspidal Pairs and Affine Springer Fibers |
| topic | Representation Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2311.17106 |