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

Funke, Jens; Hofmann, Eric

doi:10.1090/tran/8289

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

Funke, Jens; Hofmann, Eric

Authors

Professor Jens Funke jens.funke@durham.ac.uk
Professor

Eric Hofmann

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
Related Public URLs	https://arxiv.org/abs/1903.00262