Restriction of $p$-adic representations of $\mathrm{GL}_2(\mathbf{Q}_p)$ to parahoric subgroups
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866908455011352576 |
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| author | Dotto, Andrea |
| author_facet | Dotto, Andrea |
| contents | Without using the $p$-adic Langlands correspondence, we prove that for many finite length smooth representations of $\mathrm{GL}_2(\mathbf{Q}_p)$ on $p$-torsion modules the $\mathrm{GL}_2(\mathbf{Q}_p)$-linear morphisms coincide with the morphisms that are linear for the normalizer of a parahoric subgroup. We identify this subgroup to be the Iwahori subgroup in the supersingular case, and $\mathrm{GL}_2(\mathbf{Z}_p)$ in the principal series case. As an application, we relate the action of parahoric subgroups to the action of the inertia group of $\mathrm{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_p)$, and we prove that if an irreducible Banach space representation $Π$ of $\mathrm{GL}_2(\mathbf{Q}_p)$ has infinite $\mathrm{GL}_2(\mathbf{Z}_p)$-length then a twist of $Π$ has locally algebraic vectors. This answers a question of Dospinescu. We make the simplifying assumption that $p > 3$ and that all our representations are generic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_12827 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Restriction of $p$-adic representations of $\mathrm{GL}_2(\mathbf{Q}_p)$ to parahoric subgroups Dotto, Andrea Number Theory Representation Theory Without using the $p$-adic Langlands correspondence, we prove that for many finite length smooth representations of $\mathrm{GL}_2(\mathbf{Q}_p)$ on $p$-torsion modules the $\mathrm{GL}_2(\mathbf{Q}_p)$-linear morphisms coincide with the morphisms that are linear for the normalizer of a parahoric subgroup. We identify this subgroup to be the Iwahori subgroup in the supersingular case, and $\mathrm{GL}_2(\mathbf{Z}_p)$ in the principal series case. As an application, we relate the action of parahoric subgroups to the action of the inertia group of $\mathrm{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_p)$, and we prove that if an irreducible Banach space representation $Π$ of $\mathrm{GL}_2(\mathbf{Q}_p)$ has infinite $\mathrm{GL}_2(\mathbf{Z}_p)$-length then a twist of $Π$ has locally algebraic vectors. This answers a question of Dospinescu. We make the simplifying assumption that $p > 3$ and that all our representations are generic. |
| title | Restriction of $p$-adic representations of $\mathrm{GL}_2(\mathbf{Q}_p)$ to parahoric subgroups |
| topic | Number Theory Representation Theory |
| url | https://arxiv.org/abs/2111.12827 |