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

TL;DR

This paper proves that supercuspidal representations of GL(n) over a non-archimedean field are distinguished by a unitary involution if and only if they are Galois invariant, extending known results to a local setting and modular cases.

Contribution

It provides a local proof for the Galois invariance criterion for distinguished supercuspidal representations, applicable to complex and l-modular cases, generalizing previous global results.

Findings

Representation is distinguished iff Galois invariant.

Dimension of Hom space is at most one.

Applicable to both complex and modular representations.

Abstract

Let $F/F_{0}$ be a quadratic extension of non-archimedean locally compact fields of residue characteristic $p\neq 2$. 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 $l\neq p$. We further study the dimension of $\mathrm{Hom}_{G^{\tau}}(\pi,1)$ and show that it is at most one.