Cuspidal endo-support and strong beta extensions
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866911562793484288 |
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| author | Helm, David Kurinczuk, Robert Skodlerack, Daniel Stevens, Shaun |
| author_facet | Helm, David Kurinczuk, Robert Skodlerack, Daniel Stevens, Shaun |
| contents | Let $G$ be an inner form of a general linear group or classical group over a non-archimedean local field of residual characteristic $p$, assumed odd in the classical case. We prove that every smooth representation of $G$ over an algebraically closed field $R$ of characteristic $\ell\neq p$ contains a maximal semisimple character, i.e., one for which the point in the building of the corresponding centralizer is a vertex. Further, for every endo-parameter adapted to $G$, we define its support, which leads also to the notion of cuspidal endo-support of an irreducible representation, and we relate this to its cuspidal support. We also introduce beta extensions for strong facets in the building of a centralizer, and show these are sufficient for the construction of types. These results are used in a subsequent paper to decompose the category of smooth $R$-representations of $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_01781 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Cuspidal endo-support and strong beta extensions Helm, David Kurinczuk, Robert Skodlerack, Daniel Stevens, Shaun Representation Theory Let $G$ be an inner form of a general linear group or classical group over a non-archimedean local field of residual characteristic $p$, assumed odd in the classical case. We prove that every smooth representation of $G$ over an algebraically closed field $R$ of characteristic $\ell\neq p$ contains a maximal semisimple character, i.e., one for which the point in the building of the corresponding centralizer is a vertex. Further, for every endo-parameter adapted to $G$, we define its support, which leads also to the notion of cuspidal endo-support of an irreducible representation, and we relate this to its cuspidal support. We also introduce beta extensions for strong facets in the building of a centralizer, and show these are sufficient for the construction of types. These results are used in a subsequent paper to decompose the category of smooth $R$-representations of $G$. |
| title | Cuspidal endo-support and strong beta extensions |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2604.01781 |