A Jacquet-Langlands functor for $p$-adic locally analytic representations
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910206158438400 |
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| author | Dospinescu, Gabriel Camargo, Juan Esteban Rodríguez |
| author_facet | Dospinescu, Gabriel Camargo, Juan Esteban Rodríguez |
| contents | We study the locally analytic theory of infinite level local Shimura varieties. As a main result, we prove that in the case of a duality of local Shimura varieties, the locally analytic vectors of different period sheaves at infinite level are independent of the actions of the $p$-adic Lie groups $G$ and $G_b$ of the two towers; this generalizes a result of Pan for the Lubin-Tate and Drinfeld spaces for $\mathrm{GL}_2$. We apply this theory to show that Scholze's $p$-adic Jacquet-Langlands functor commutes with the passage to locally analytic vectors, and is compatible with central characters of Lie algebras. We also prove that the compactly supported de Rham cohomology of the two towers are isomorphic as smooth representations of $G\times G_b$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_17082 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Jacquet-Langlands functor for $p$-adic locally analytic representations Dospinescu, Gabriel Camargo, Juan Esteban Rodríguez Number Theory Representation Theory We study the locally analytic theory of infinite level local Shimura varieties. As a main result, we prove that in the case of a duality of local Shimura varieties, the locally analytic vectors of different period sheaves at infinite level are independent of the actions of the $p$-adic Lie groups $G$ and $G_b$ of the two towers; this generalizes a result of Pan for the Lubin-Tate and Drinfeld spaces for $\mathrm{GL}_2$. We apply this theory to show that Scholze's $p$-adic Jacquet-Langlands functor commutes with the passage to locally analytic vectors, and is compatible with central characters of Lie algebras. We also prove that the compactly supported de Rham cohomology of the two towers are isomorphic as smooth representations of $G\times G_b$. |
| title | A Jacquet-Langlands functor for $p$-adic locally analytic representations |
| topic | Number Theory Representation Theory |
| url | https://arxiv.org/abs/2411.17082 |