Modulo \(\ell\) 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 \(\overline{\mathbb{F}}_{\ell}\)-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 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 \(\mathrm{GL}_2(F)\)-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 \(\mathrm{SL}_2(E)\).