Symplectic models for Unitary groups

TL;DR

This paper investigates the distinction of representations of quasi-split unitary groups by symplectic groups, proving non-existence of certain cuspidal representations and classifying others using theta correspondence, with conjectures for a complete classification.

Contribution

It establishes the non-existence of cuspidal distinguished representations and classifies those with symplectic periods in four-variable cases, proposing a conjectural classification for all such representations.

Findings

No cuspidal representations of $U_{2n}(F)$ are distinguished by $Sp_{2n}(F)$ over non-archimedean fields.

Complete classification of certain distinguished representations in four variables.

Global non-existence of cuspidal representations with nonzero symplectic period.

Abstract

In analogy with the study of representations of $GL_{2n}(F)$ distinguished by $Sp_{2n}(F)$, where $F$ is a local field, in this paper we study representations of $U_{2n}(F)$ distinguished by $Sp_{2n}(F)$. (Only quasi-split unitary groups are considered in this paper since they are the only ones which contain $Sp_{2n}(F)$.) We prove that there are no cuspidal representations of $U_{2n}(F)$ distinguished by $Sp_{2n}(F)$ for $F$ a non-archimedean local field. We also prove the corresponding global theorem that there are no cuspidal representations of $U_{2n}({\mathbb A}_k)$ with nonzero period integral on $Sp_{2n}(k) \backslash Sp_{2n}({\mathbb A}_k)$ for $k$ any number field or a function field. We completely classify representations of quasi-split unitary group in four variables over local and global fields with nontrivial symplectic periods using methods of theta correspondence. We propose a conjectural answer for the classification of all representations of a quasi-split unitary group distinguished by $Sp_{2n}(F)$.