Symplectic models for Unitary groups

Abstract

In analogy with the study of representations of $G{L}_{2n}(F)$ distinguished by $S{p}_{2n}(F)$, where $F$ is a local field, in this paper we study representations of $U_{2n}(F)$ distinguished by $S{p}_{2n}(F)$. (Only quasi-split unitary groups are considered in this paper since they are the only ones which contain $S{p}_{2n}(F)$.) We prove that there are no cuspidal representations of $U_{2n}(F)$ distinguished by $S{p}_{2n}(F)$ 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 $S{p}_{2n}(k)\backslash S{p}_{2n}({\mathbb{A}}_k)$ 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 $S{p}_{2n}(F)$.

minor changes; to appear in the Transactions of the AMS