On tori periods of Weil representations of unitary groups

Abstract

We determine the restriction of Weil representations of unitary groups to maximal tori. In the local case, we show that the Weil representation contains a pair of compatible characters if and only if a root number condition holds. In the global case, we show that a torus period corresponding to a maximal anisotropic torus of the global theta lift of a character does not vanish if and only if the local condition is satisfied everywhere and a central value of an L-function does not vanish. Our proof makes use of the seesaw argument and of the well-known theta lifting results from to U(1) to U(1). Our results are used in [1, 2] to construct Arthur packets for G2.

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Item Type:	Article
Authors/Creators:	Borade, N. Franzel, J. Girsch, J. Yao, W. Yu, Q. Zelingher, E.
Copyright, Publisher and Additional Information:	© The Author(s) 2025. Open Access: This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Keywords:	Theta correspondence; Periods; Special values of L-functions; Gan–Gross–Prasad conjectures
Dates:	Accepted: 14 February 2025 Published (online): 21 May 2025 Published: July 2025
Institution:	The University of Sheffield
Academic Units:	The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematical and Physical Sciences
Depositing User:	Symplectic Sheffield
Date Deposited:	28 May 2025 07:20
Last Modified:	28 May 2025 07:20
Status:	Published
Publisher:	Springer Science and Business Media LLC
Refereed:	Yes
Identification Number:	10.1007/s00029-025-01047-4
Open Archives Initiative ID (OAI ID):	oai:eprints.whiterose.ac.uk:227128

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On tori periods of Weil representations of unitary groups

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