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

Authors: Jens Funke and Eric Hofmann
Journal: Trans. Amer. Math. Soc. 374 (2021), 2909-2947
MSC (2020): Primary 11F27; Secondary 11F55, 14C25
DOI: https://doi.org/10.1090/tran/8289
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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