Jacquet modules of Tate cohomology and base change lifting
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arXiv
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| Format: | Preprint |
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2025
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| author | Dhar, Sabyasachi Nadimpalli, Santosh |
| author_facet | Dhar, Sabyasachi Nadimpalli, Santosh |
| contents | Let $G$ be a connected reductive group defined over a
non-Archimedean local field $F$ of residue characteristic $p$. Let
$\ell$ be a prime number distinct from $p$. Let $E$ be a cyclic
Galois extension of $F$ with $[E:F]=\ell$. Let $Π$ be a finite
length $\overline{\mathbb{F}}_\ell$-representation (or an
$\ell$-modular representation) of $G(E)\rtimes {\rm Gal}(E/F)$. In
this context, we prove a conjecture of Treumann and Venkatesh which
predicts that the Tate cohomology groups
$\widehat{H}^i({\rm Gal}(E/F), Π)$ are finite length
representations of $G(F)$. We discuss the explicit computation of
these Tate cohomology groups when $G$ is ${\rm GL}_n$ and $Π$ is
obtained as a base change lifting of a depth-zero cuspidal
representation of ${\rm GL}_n(F)$. The primary novelty from our
previous work is that we treat the case where $Π$ is possibly
non-cuspidal. We also study the
${\rm Gal}(\mathbb{F}_{q^\ell}/\mathbb{F}_q)$-Tate cohomology groups
of the mod-$\ell$ reduction of the unipotent cuspidal representation
of ${\rm Sp}_4(\mathbb{F}_{q^\ell})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_09773 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Jacquet modules of Tate cohomology and base change lifting Dhar, Sabyasachi Nadimpalli, Santosh Representation Theory Let $G$ be a connected reductive group defined over a non-Archimedean local field $F$ of residue characteristic $p$. Let $\ell$ be a prime number distinct from $p$. Let $E$ be a cyclic Galois extension of $F$ with $[E:F]=\ell$. Let $Π$ be a finite length $\overline{\mathbb{F}}_\ell$-representation (or an $\ell$-modular representation) of $G(E)\rtimes {\rm Gal}(E/F)$. In this context, we prove a conjecture of Treumann and Venkatesh which predicts that the Tate cohomology groups $\widehat{H}^i({\rm Gal}(E/F), Π)$ are finite length representations of $G(F)$. We discuss the explicit computation of these Tate cohomology groups when $G$ is ${\rm GL}_n$ and $Π$ is obtained as a base change lifting of a depth-zero cuspidal representation of ${\rm GL}_n(F)$. The primary novelty from our previous work is that we treat the case where $Π$ is possibly non-cuspidal. We also study the ${\rm Gal}(\mathbb{F}_{q^\ell}/\mathbb{F}_q)$-Tate cohomology groups of the mod-$\ell$ reduction of the unipotent cuspidal representation of ${\rm Sp}_4(\mathbb{F}_{q^\ell})$. |
| title | Jacquet modules of Tate cohomology and base change lifting |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2507.09773 |