SYMPLECTIC MODELS FOR UNITARY GROUPS.

In analogy with the study of representations of GL2n(F) distinguished by Sp2n(F), where F is a local field, we study representations of U2n(F) distinguished by Sp2n(F) in this paper. (Only quasisplit unitary groups are considered in this paper since they are the only ones which contain Sp2n(F).) We prove that there are no cuspidal representations of U2n(F) distinguished by Sp2n(F) for F a nonarchimedean local field. We also prove the corresponding global theorem that there are no cuspidal automorphic representations of U2n(Ak) with nonzero period integral on Sp2n(k)\ Sp2n(Ak) for k any number field or a function field. We completely classify representations of quasisplit unitary groups 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 quasisplit unitary group distinguished by Sp2n(F). [ABSTRACT FROM AUTHOR]