Modulo <i>ℓ</i> distinction problems

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 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 $\operatorname {GL}_2(F)$ -distinguished representations of $\operatorname {GL}_2(E)$ . Using this classification we discuss a modular version of the Prasad conjecture for $\operatorname {PGL}_2$ . 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 $\operatorname {SL}_2(F)$ -distinguished modular representations of $\operatorname {SL}_2(E)$ .