Modulo $\ell$ distinction problems

TL;DR

This paper advances the understanding of modular representations of reductive groups over local fields, proving lifting theorems, verifying conjectures in the modular setting, and classifying distinguished representations, with implications for the Prasad conjecture.

Contribution

It introduces a lifting theorem for supercuspidal modular representations and verifies the Jacquet conjecture in the modular context, extending prior results to characteristic 2.

Findings

Lifting supercuspidal representations from modular to characteristic zero.

Verification of the Jacquet conjecture for modular representations with irreducible, conjugate-self-dual parameters.

Complete classification of GL₂(F)-distinguished representations of GL₂(E).

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 $\overline{\mathbb{F}}_{\ell}$-representations of $\mathrm{GL}_n(F)$ distinguished by an arbitrary closed subgroup $H$ to a distinguished supercuspidal $\overline{\mathbb{Q}}_{\ell}$-representation. Given a quadratic field extension $E/F$ and an irreducible $\overline{\mathbb{F}}_{\ell}$-representation $\pi$ of $\mathrm{GL}_n(E)$, we verify the Jacquet conjecture in the modular setting that if the Langlands parameter $\phi_\pi$ is irreducible and conjugate-self-dual, then $\pi$ is either $\mathrm{GL}_n(F)$-distinguished or $(\mathrm{GL}_n(F),\omega_{E/F})$-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 S\'echerre to the case $p=2$. We give another application of our lifting theorem for supercuspidal representations distinguished by a unitary involution, extending one result of Zou to $p=2$. After that, we give a complete classification of the $\mathrm{GL}_2(F)$-distinguished representations of $\mathrm{GL}_2(E)$. Using this classification we discuss a modular version of the Prasad conjecture for $\mathrm{PGL}_2$. We show that the "classical" Prasad conjecture fails in the modular setting. We propose a solution using non-nilpotent Weil-Deligne representations. Finally, we apply the restriction method of Anandavardhanan and Prasad to classify the $\mathrm{SL}_2(F)$-distinguished modular representations of $\mathrm{SL}_2(E)$.