On freeness of compactly induced mod-$p$ representations of $\rm{SL}_{2}(F)$
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908600664850432 |
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| author | Das, Arpan |
| author_facet | Das, Arpan |
| contents | Let $p$ be a prime, and $F$ a non-archimedean local field with residue characteristic $p$ and ring of integers $\mathcal{O}_{F}$. Set $G_{S}:={\rm SL}_{2}(F)$and $K_{0}:={\rm SL}_{2}(\mathcal{O}_{F})$ . For a smooth irreducible $\bar{\mathbb{F}}_{p}$-representation $σ$ of $K_{0}$, we study the structure of the compact induction ${\rm ind}_{K_{0}}^{G_{S}}(σ)$ as a left module over the standard spherical Hecke algebra ${\rm End}_{G_{S}}\left({\rm ind}_{K_{0}}^{G_{S}}(σ)\right)$. We prove that it is free and of infinite rank. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_16324 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On freeness of compactly induced mod-$p$ representations of $\rm{SL}_{2}(F)$ Das, Arpan Representation Theory Let $p$ be a prime, and $F$ a non-archimedean local field with residue characteristic $p$ and ring of integers $\mathcal{O}_{F}$. Set $G_{S}:={\rm SL}_{2}(F)$and $K_{0}:={\rm SL}_{2}(\mathcal{O}_{F})$ . For a smooth irreducible $\bar{\mathbb{F}}_{p}$-representation $σ$ of $K_{0}$, we study the structure of the compact induction ${\rm ind}_{K_{0}}^{G_{S}}(σ)$ as a left module over the standard spherical Hecke algebra ${\rm End}_{G_{S}}\left({\rm ind}_{K_{0}}^{G_{S}}(σ)\right)$. We prove that it is free and of infinite rank. |
| title | On freeness of compactly induced mod-$p$ representations of $\rm{SL}_{2}(F)$ |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2510.16324 |