Irreducible smooth representations in defining characteristic without central character
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866913968775233536 |
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| author | Le, Daniel |
| author_facet | Le, Daniel |
| contents | Let $p>3$ be a prime, $n>1$ be an integer, and $F$ be a non-archimedean local field with residue field a proper finite extension of $\mathbb{F}_p$. Let $E$ be an algebraically closed countable field extension of the residue field of $F$. In this short note, we explain how the methods from arXiv:1809.10247 and arXiv:2210.07281 can be used to construct irreducible smooth representations of $\mathrm{GL}_n(F)$ over $E$ without a central character. We also construct irreducible smooth representations of $\mathrm{GL}_n(F)$ over $E$ with simultaneously a central character, nonscalar endomorphisms, and if $n>3$, without a Hecke eigenvalue. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_01766 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Irreducible smooth representations in defining characteristic without central character Le, Daniel Representation Theory Number Theory 22E50 Let $p>3$ be a prime, $n>1$ be an integer, and $F$ be a non-archimedean local field with residue field a proper finite extension of $\mathbb{F}_p$. Let $E$ be an algebraically closed countable field extension of the residue field of $F$. In this short note, we explain how the methods from arXiv:1809.10247 and arXiv:2210.07281 can be used to construct irreducible smooth representations of $\mathrm{GL}_n(F)$ over $E$ without a central character. We also construct irreducible smooth representations of $\mathrm{GL}_n(F)$ over $E$ with simultaneously a central character, nonscalar endomorphisms, and if $n>3$, without a Hecke eigenvalue. |
| title | Irreducible smooth representations in defining characteristic without central character |
| topic | Representation Theory Number Theory 22E50 |
| url | https://arxiv.org/abs/2407.01766 |