Cuspidal $\ell$-modular representations of ${\mathrm{GL}}_n(F)$ distinguished by a Galois involution

Abstract

Let $F/F_0$ be a quadratic extension of non-Archimedean locally compact fields of residual characteristic $p\ne 2$ with Galois automorphism $\sigma$, and let $R$ be an algebraically closed field of characteristic $\ell \notin \{0,p\}$. We reduce the classification of ${\mathrm{GL}}_n(F_0)$-distinguished cuspidal $R$-representations of ${\mathrm{GL}}_n(F)$ to the level $0$ setting. Moreover, under a parity condition, we give necessary conditions for a $\sigma$-selfdual cuspidal $R$-representation to be distinguished. Finally, we classify the distinguished cuspidal ${\overline{\mathbb{F}}}_{\ell }$-representations of ${\mathrm{GL}}_n(F)$ having a distinguished cuspidal lift to ${\overline{\mathbb{Q}}}_{\ell }$.

42 pages