On modulo $\ell$ cohomology of $p$-adic Deligne-Lusztig varieties for $GL_n$
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910639957475328 |
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| author | Löwit, Jakub |
| author_facet | Löwit, Jakub |
| contents | In 1976, Deligne and Lusztig realized the representation theory of finite groups of Lie type inside étale cohomology of certain algebraic varieties. Recently, a $p$-adic version of this theory started to emerge: there are $p$-adic Deligne-Lusztig spaces, whose cohomology encodes representation theoretic information for $p$-adic groups - for instance, it partially realizes local Langlands correspondence with characteristic zero coefficients. However, the parallel case of coefficients of positive characteristic $\ell \neq p$ has not been inspected so far. The purpose of this article is to initiate such an inspection. In particular, we relate cohomology of certain $p$-adic Deligne-Lusztig spaces to Vignéras's modular local Langlands correspondence for $\mathbf{GL}_n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_11176 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On modulo $\ell$ cohomology of $p$-adic Deligne-Lusztig varieties for $GL_n$ Löwit, Jakub Representation Theory Algebraic Geometry In 1976, Deligne and Lusztig realized the representation theory of finite groups of Lie type inside étale cohomology of certain algebraic varieties. Recently, a $p$-adic version of this theory started to emerge: there are $p$-adic Deligne-Lusztig spaces, whose cohomology encodes representation theoretic information for $p$-adic groups - for instance, it partially realizes local Langlands correspondence with characteristic zero coefficients. However, the parallel case of coefficients of positive characteristic $\ell \neq p$ has not been inspected so far. The purpose of this article is to initiate such an inspection. In particular, we relate cohomology of certain $p$-adic Deligne-Lusztig spaces to Vignéras's modular local Langlands correspondence for $\mathbf{GL}_n$. |
| title | On modulo $\ell$ cohomology of $p$-adic Deligne-Lusztig varieties for $GL_n$ |
| topic | Representation Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2404.11176 |