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

Abstract

Let $F/F_0$ be a quadratic extension of non-archimedean locally compact fields of residue characteristic $p\ne 2$. Let $R$ be an algebraically closed field of characteristic different from $p$. For $\pi$ a supercuspidal representation of $G={\mathrm{GL}}_n(F)$ over $R$ and $G^{\tau }$ a unitary group in $n$ variables contained in $G$, we prove that $\pi$ is distinguished by $G^{\tau }$ if and only if $\pi$ is Galois invariant. When $R=\mathbb{C}$ and $F$ is a $p$-adic field, this result first as a conjecture proposed by Jacquet was proved in 2010's by Feigon-Lapid-Offen by using global method. Our proof is local which works for both complex case and $l$-modular case with $l\ne p$. We further study the dimension of ${\mathrm{Hom}}_{G^{\tau }}(\pi ,1)$ and show that it is at most one.

Published version in BSMF with minor changes

References in corpus (3)

• The local Jacquet--Langlands correspondence and congruences modulo {\ell}
• Galois self-dual cuspidal types and Asai local factors
• Test vectors for finite periods and base change

Cited by in corpus (1)

• Supercuspidal representations of ${\mathrm{GL}}_n(F)$ distinguished by an orthogonal involution