Symplectic models for unitary groups

Dijols, Sarah; Prasad, Dipendra

doi:10.1090/tran/7651

Symplectic models for unitary groups
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by Sarah Dijols and Dipendra Prasad PDF

Trans. Amer. Math. Soc. 372 (2019), 1833-1866 Request permission

Abstract:

In analogy with the study of representations of $\operatorname {GL}_{2n}(F)$ distinguished by $\operatorname {Sp}_{2n}(F)$, where $F$ is a local field, we study representations of $\operatorname {U}_{2n}(F)$ distinguished by $\operatorname {Sp}_{2n}(F)$ in this paper. (Only quasisplit unitary groups are considered in this paper since they are the only ones which contain $\operatorname {Sp}_{2n}(F)$.) We prove that there are no cuspidal representations of $\operatorname {U}_{2n}(F)$ distinguished by $\operatorname {Sp}_{2n}(F)$ for $F$ a nonarchimedean local field. We also prove the corresponding global theorem that there are no cuspidal automorphic representations of $\operatorname {U}_{2n}(\mathbb {A}_k)$ with nonzero period integral on $\operatorname {Sp}_{2n}(k) \backslash \operatorname {Sp}_{2n}(\mathbb {A}_k)$ 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 $\operatorname {Sp}_{2n}(F)$.

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