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

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- by Jens Funke and Eric Hofmann PDF
- Trans. Amer. Math. Soc.
**374**(2021), 2909-2947 Request permission

## 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.## References

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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
- Email: jens.funke@durham.ac.uk
**Eric Hofmann**- Affiliation: Mathematisches Institut, Universität Heidelberg, Im Neuenheimer Feld 205, D-69120 Heidelberg, Germany
- MR Author ID: 1043940
- ORCID: 0000-0003-4617-5765
- Email: hofmann@mathi.uni-heidelberg.de
- 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
- © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**374**(2021), 2909-2947 - MSC (2020): Primary 11F27; Secondary 11F55, 14C25
- DOI: https://doi.org/10.1090/tran/8289
- MathSciNet review: 4223037