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

Abstract

Let $F/F_0$ be a quadratic extension of non-Archimedean locally compact fields with residual characteristic $p\ne 2$, and $\ell$ be a prime number different from $p$. We classify those $\ell$-modular cuspidal irreducible representations of ${\mathrm{GL}}_n(F)$ which are ${\mathrm{GL}}_n(F_0)$-distinguished, that is, which carry a non-zero ${\mathrm{GL}}_n(F_0)$-invariant linear form. In the case when $\ell \ne 2$, an $\ell$-modular cuspidal representation of ${\mathrm{GL}}_n(F)$ is ${\mathrm{GL}}_n(F_0)$-distinguished if and only if it lifts to a ${\mathrm{GL}}_n(F_0)$-distinguished cuspidal $\ell$-adic representation, whereas when $\ell =2$, it is ${\mathrm{GL}}_n(F_0)$-distinguished if and only if it is conjugate-self-dual.