Year: 2024
Type: article
Abstract: Let $F$ be a non-archimedean local field of characteristic different from 2 and residual characteristic $p$ . This paper concerns the $\ell$ -modular representations of a connected reductive group $G$ distinguished by a Galois involution, with $\ell$ an odd prime different from $p$ . We start by proving a general theorem allowing to lift supercuspidal $\overline {\mathbf {F}}_{\ell }$ -representations of $\operatorname {GL}_n(F)$ distinguished by an arbitrary closed subgroup $H$ to a distinguished supercuspidal $\overline {\mathbf {Q}}_{\ell }$ -representation. Given a quadratic field extension $E/F$ and an irreducible $\overline {\mathbf {F}}_{\ell }$ -representation $\pi$ of $\operatorname {GL}_n(E)$ , we verify the Jacquet conjecture in the modular setting that if the Langlands parameter $\phi _\pi$ is irreducible and conjugate-selfdual, then $\pi$ is either $\operatorname {GL}_n(F)$ -distinguished or $(\operatorname {GL}_{n}(F),\omega _{E/F})$ -distinguished (where $\omega _{E/F}$ is the quadratic character of $F^\times$ associated to the quadratic field extension $E/F$ by the local class field theory), but not both, which extends one result of Sécherre to the case $p=2$ . We give another application
Source: Compositio Mathematica
Authors: Peiyi Cui, Thomas Lanard, Hengfei Lu
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4402994097