The construction of Green currents and singular theta lifts for unitary groups

Funke, Jens; Hofmann, Eric

Abstract

With applications in the Kudla program in mind we employ singular theta lifts for the reductive dual pair $ \mathrm {U}(p,q)\times \mathrm {U}(1,1)$ to construct two different kinds of Green forms for codimension $ q$-cycles in Shimura varieties associated to unitary groups. We establish an adjointness result between our singular theta lift and the Kudla-Millson lift. Further, we compare the two Greens forms and obtain modularity for the generating function of the difference of the two Green forms. Finally, we show that the Green forms obtained by the singular theta lift satisfy an eigenvalue equation for the Laplace operator and conclude that our Green forms coincide with the ones constructed by Oda and Tsuzuki by different means.

Citation

Funke, J., & Hofmann, E. (2021). The construction of Green currents and singular theta lifts for unitary groups. Transactions of the American Mathematical Society, 374(4), 2909-2947. https://doi.org/10.1090/tran/8289

• Journal Article Type: Article
• Acceptance Date: Aug 25, 2020
• Online Publication Date: Jan 27, 2021
• Publication Date: 2021
• Deposit Date: Sep 1, 2020
• Publicly Available Date: Sep 16, 2020
• Journal: Transactions of the American Mathematical Society
• Print ISSN: 0002-9947
• Electronic ISSN: 1088-6850
• Publisher: American Mathematical Society
• Peer Reviewed: Peer Reviewed
• Volume: 374
• Issue: 4
• Pages: 2909-2947
• DOI: https://doi.org/10.1090/tran/8289
• Public URL: https://durham-repository.worktribe.com/output/1293443
• Related Public URLs: https://arxiv.org/abs/1903.00262