Modulo \(\ell\) distinction problems (Q6622800)

Let \(F\) denote a non-Archimedean local field of characteristic different from \(2\) and residual characteristic \(p\). Let \(G\) stand for \(F\)-points of a connected reductive group over \(F\). In the paper under review, the authors study distinguished \(l\)-modular representations of \(G\) for an odd prime \(l\) different from \(p\).\N\NThe authors first prove that for a supercuspidal \(H\)-distinguished \(\overline{F}_l\)-representation of \(\mathrm{GL}_n(F)\), \(H\) being a closed subgroup of \(\mathrm{GL}_n(F)\), there exists a supercuspidal \(H\)-distinguished \(\overline{\mathbb{Q}}_l\)-lift. This directly implies that there is no supercuspidal \(\overline{F}_l\)-representation of \(\mathrm{GL}_n(F)\) distinguished by \(\mathrm{Sp}_{2n}(F)\).\N\NTwo further applications are given. The first is a modular version of the Jacquet conjecture, which states that if the Langlands parameter of a representation is irreducible and conjugate-selfdual, then the representation is either \(\mathrm{GL}_n(F)\)-distinguished or \((\mathrm{GL}_n(F), \omega_{E / F})\)-distinguished, \(\omega_{E/F}\) being a quadratic character obtained by the local class field theory for a quadratic field extension \(E\) of \(F\). The second application concerns supercuspidal representations distinguished by a unitary involution.\N\NA complete classification of the \(\mathrm{GL}_2(F)\)-distinguished representations of \(\mathrm{GL}_2(E)\) is also provided, which enables the authors to discuss a modular version of the Prasad conjecture for \(\mathrm{PGL}_2\).