Reduction to depth zero for tame p-adic groups via Hecke algebra isomorphisms
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866910566112559104 |
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| author | Adler, Jeffrey D. Fintzen, Jessica Mishra, Manish Ohara, Kazuma |
| author_facet | Adler, Jeffrey D. Fintzen, Jessica Mishra, Manish Ohara, Kazuma |
| contents | Let $F$ be a nonarchimedean local field of residual characteristic $p$. Let $G$ denote a connected reductive group over $F$ that splits over a tamely ramified extension of $F$. Let $(K ,ρ)$ be a type as constructed by Kim and Yu. We show that there exists a twisted Levi subgroup $G^0 \subset G$ and a type $(K^0, ρ^0)$ for $G^0$ such that the corresponding Hecke algebras $\mathcal{H}(G(F), (K, ρ))$ and $\mathcal{H}(G^0(F), (K^0, ρ^0))$ are isomorphic. If $p$ does not divide the order of the absolute Weyl group of $G$, then every Bernstein block is equivalent to modules over such a Hecke algebra. Hence, under this assumption on $p$, our result implies that every Bernstein block is equivalent to a depth-zero Bernstein block. This allows one to reduce many problems about (the category of) smooth, complex representations of $p$-adic groups to analogous problems about (the category of) depth-zero representations.
Our isomorphism of Hecke algebras is very explicit and also includes an explicit description of the Hecke algebras as semi-direct products of an affine Hecke with a twisted group algebra. Moreover, we work with arbitrary algebraically closed fields of characteristic different from $p$ as our coefficient field.
This paper relies on a prior axiomatic result about the structure of Hecke algebras by the same authors and a key ingredient consists of extending the quadratic character of Fintzen--Kaletha--Spice to the support of the Hecke algebra, which might be of independent interest. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_07805 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Reduction to depth zero for tame p-adic groups via Hecke algebra isomorphisms Adler, Jeffrey D. Fintzen, Jessica Mishra, Manish Ohara, Kazuma Representation Theory Number Theory Let $F$ be a nonarchimedean local field of residual characteristic $p$. Let $G$ denote a connected reductive group over $F$ that splits over a tamely ramified extension of $F$. Let $(K ,ρ)$ be a type as constructed by Kim and Yu. We show that there exists a twisted Levi subgroup $G^0 \subset G$ and a type $(K^0, ρ^0)$ for $G^0$ such that the corresponding Hecke algebras $\mathcal{H}(G(F), (K, ρ))$ and $\mathcal{H}(G^0(F), (K^0, ρ^0))$ are isomorphic. If $p$ does not divide the order of the absolute Weyl group of $G$, then every Bernstein block is equivalent to modules over such a Hecke algebra. Hence, under this assumption on $p$, our result implies that every Bernstein block is equivalent to a depth-zero Bernstein block. This allows one to reduce many problems about (the category of) smooth, complex representations of $p$-adic groups to analogous problems about (the category of) depth-zero representations. Our isomorphism of Hecke algebras is very explicit and also includes an explicit description of the Hecke algebras as semi-direct products of an affine Hecke with a twisted group algebra. Moreover, we work with arbitrary algebraically closed fields of characteristic different from $p$ as our coefficient field. This paper relies on a prior axiomatic result about the structure of Hecke algebras by the same authors and a key ingredient consists of extending the quadratic character of Fintzen--Kaletha--Spice to the support of the Hecke algebra, which might be of independent interest. |
| title | Reduction to depth zero for tame p-adic groups via Hecke algebra isomorphisms |
| topic | Representation Theory Number Theory |
| url | https://arxiv.org/abs/2408.07805 |