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

TL;DR

This paper classifies cuspidal mbda-modular representations of GL_n(F) distinguished by a Galois involution, revealing conditions for distinction based on the prime e5 and lifting properties.

Contribution

It provides a complete classification of e5-modular cuspidal representations distinguished by GL_n(F_0), linking distinction to lifting and conjugate-self-duality.

Findings

For e5e9 2, distinction is characterized by conjugate-self-duality.

For e5e9 not 2, distinction corresponds to lifting from e5-adic representations.

The classification depends on the prime e5 and the representation's relation to e5-adic counterparts.

Abstract

Let $F/F_0$ be a quadratic extension of non-Archimedean locally compact fields with residual characteristic $p\neq2$, and $\ell$ be a prime number different from $p$. We classify those $\ell$-modular cuspidal irreducible representations of ${\rm GL}_n(F)$ which are ${\rm GL}_n(F_0)$-distinguished, that is, which carry a non-zero ${\rm GL}_n(F_0)$-invariant linear form. In the case when $\ell\neq2$, an $\ell$-modular cuspidal representation of ${\rm GL}_n(F)$ is ${\rm GL}_n(F_0)$-distinguished if and only if it lifts to a ${\rm GL}_n(F_0)$-distinguished cuspidal $\ell$-adic representation, whereas when $\ell=2$, it is ${\rm GL}_n(F_0)$-distinguished if and only if it is conjugate-self-dual.