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Transactions of the American Mathematical Society

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



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

Authors: Jens Funke and Eric Hofmann
Journal: Trans. Amer. Math. Soc. 374 (2021), 2909-2947
MSC (2020): Primary 11F27; Secondary 11F55, 14C25
Published electronically: January 27, 2021
MathSciNet review: 4223037
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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.

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Additional Information

Jens Funke
Affiliation: Department of Mathematical Sciences, University of Durham, South Road, Durham DH1 3LE, United Kingdom
MR Author ID: 652687
ORCID: 0000-0003-2694-4539

Eric Hofmann
Affiliation: Mathematisches Institut, Universität Heidelberg, Im Neuenheimer Feld 205, D-69120 Heidelberg, Germany
MR Author ID: 1043940
ORCID: 0000-0003-4617-5765

Received by editor(s): September 23, 2019
Received by editor(s) in revised form: February 26, 2020, and August 25, 2020
Published electronically: January 27, 2021
Additional Notes: The second author was supported by a research fellowship (Forschungsstipendium) of the DFG number HO 6123/1-1
Article copyright: © Copyright 2021 American Mathematical Society