On ${\rm Sp}$-distinguished representations of the quasi-split unitary groups

TL;DR

This paper investigates the distinction of representations of quasi-split unitary groups with respect to symplectic groups, verifying a conjecture for many cases and exploring the transfer of distinguished properties under base change.

Contribution

It confirms the Dijols-Prasad conjecture for a broad class of discrete series representations and analyzes the transfer of distinguished representations under stable base change.

Findings

No tempered representation is distinguished in the studied family.

Identified $L$-packets with no distinguished members.

Established transfer of distinguished representations to ${ m GL}_{2n}(E)$.

Abstract

We study ${\rm Sp}_{2n}(F)$-distinction for representations of the quasi-split unitary group $U_{2n}(E/F)$ in $2n$ variables with respect to a quadratic extension $E/F$ of $p$-adic fields. A conjecture of Dijols and Prasad predicts that no tempered representation is distinguished. We verify this for a large family of representations in terms of the Moeglin-Tadic classification of the discrete series. We further study distinction for some families of non-tempered representations. In particular, we exhibit $L$-packets with no distinguished members that transfer under stable base change to ${\rm Sp}_{2n}(E)$-distinguished representations of ${\rm GL}_{2n}(E)$.