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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2510.07845 |
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| _version_ | 1866914082430386176 |
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| author | Bhattacharya, Sarbartha Chen, Tsao-Hsien |
| author_facet | Bhattacharya, Sarbartha Chen, Tsao-Hsien |
| contents | Let $G$ be a split connected reductive over a non-archimedean local field $k$. In this paper we give a description of the depth-$r$ Bernstein center of $G(k)$ for rational depths as a limit of depth-$r$ standard parahoric Hecke algebras, extending our previous work in the integral depths case (arXiv:2407.15128). Using this description, we construct maps from the space of stable functions on depth-$r$ Moy-Prasad quotients to the depth-$r$ center, and attach depth-$r$ Deligne-Lusztig parameters to smooth irreducible representations, with the assignment of parameters to irreducible representations shown to be consistent with restricted Langlands parameters for Moy-Prasad types described Chen-Debacker-Tsai (arXiv:2509.07780). As an application, we give a decomposition of the category of smooth representations into a product of full subcategories indexed by restricted depth-$r$ Langlands parameters. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_07845 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A description of the depth-$r$ Bernstein center for rational depths Bhattacharya, Sarbartha Chen, Tsao-Hsien Representation Theory Number Theory 20G25, 22E50 Let $G$ be a split connected reductive over a non-archimedean local field $k$. In this paper we give a description of the depth-$r$ Bernstein center of $G(k)$ for rational depths as a limit of depth-$r$ standard parahoric Hecke algebras, extending our previous work in the integral depths case (arXiv:2407.15128). Using this description, we construct maps from the space of stable functions on depth-$r$ Moy-Prasad quotients to the depth-$r$ center, and attach depth-$r$ Deligne-Lusztig parameters to smooth irreducible representations, with the assignment of parameters to irreducible representations shown to be consistent with restricted Langlands parameters for Moy-Prasad types described Chen-Debacker-Tsai (arXiv:2509.07780). As an application, we give a decomposition of the category of smooth representations into a product of full subcategories indexed by restricted depth-$r$ Langlands parameters. |
| title | A description of the depth-$r$ Bernstein center for rational depths |
| topic | Representation Theory Number Theory 20G25, 22E50 |
| url | https://arxiv.org/abs/2510.07845 |