Supercuspidal representations of ${\rm GL}_n({\rm F})$ distinguished by a Galois involution

Abstract

Let $F∕F_0$ be a quadratic extension of nonarchimedean locally compact fields of residual characteristic $p\ne 2$ and let $\sigma$ denote its nontrivial automorphism. Let $R$ be an algebraically closed field of characteristic different from $p$. To any cuspidal representation $\pi$ of ${GL}_n\left(F\right)$, with coefficients in $R$, such that ${\pi }^{\sigma }\simeq {\pi }^{\vee }$ (such a representation is said to be $\sigma$-selfdual) we associate a quadratic extension $D∕D_0$, where $D$ is a tamely ramified extension of $F$ and $D_0$ is a tamely ramified extension of $F_0$, together with a quadratic character of $D_0^{\times }$. When $\pi$ is supercuspidal, we give a necessary and sufficient condition, in terms of these data, for $\pi$ to be ${GL}_n\left(F_0\right)$-distinguished. When the characteristic $\ell$ of $R$ is not $2$, denoting by $\omega$ the nontrivial $R$-character of $F_0^{\times }$ trivial on $F∕F_0$-norms, we prove that any $\sigma$-selfdual supercuspidal $R$-representation is either distinguished or $\omega$-distinguished, but not both. In the modular case, that is when $\ell >0$, we give examples of $\sigma$-selfdual cuspidal nonsupercuspidal representations which are not distinguished nor $\omega$-distinguished. In the particular case where $R$ is the field of complex numbers, in which case all cuspidal representations are supercuspidal, this gives a complete distinction criterion for arbitrary complex cuspidal representations, as well as a purely local proof, for cuspidal representations, of the dichotomy and disjunction theorem due to Kable and Anandavardhanan, Kable and Tandon, when $p\ne 2$.