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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Symplectic models for unitary groups

Authors: Sarah Dijols and Dipendra Prasad
Journal: Trans. Amer. Math. Soc. 372 (2019), 1833-1866
MSC (2010): Primary 22E50
Published electronically: December 7, 2018
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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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Additional Information

Sarah Dijols
Affiliation: Aix Marseille Université, 13453, Marseille, France.

Dipendra Prasad
Affiliation: Laboratory of Modern Algebra and Applications, Saint-Petersburg State University, Russia; and Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400 005, India

Keywords: Unitary groups, symplectic models, Whittaker models, Weil representation, theta correspondence
Received by editor(s): November 5, 2016
Received by editor(s) in revised form: May 21, 2018, and May 30, 2018
Published electronically: December 7, 2018
Additional Notes: The first author thanks the ANR FERPLAY for supporting her financially during the period this work was done.
The work of the second author was supported by a grant of the Government of the Russian Federation for the state support of scientific research carried out under the supervision of leading scientists, agreement 14.W03.31.0030 dated 15.02.2018.
Article copyright: © Copyright 2018 American Mathematical Society