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Mixed Arithmetic Theta Lifting for Unitary Groups

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Relative Trace Formulas

Part of the book series: Simons Symposia ((SISY))

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Abstract

In this article, we construct mixed generating functions of special cycles and mixed arithmetic theta lifting, and prove the product relation. We then formulate a conjecture on the arithmetic inner product formula for odd rank unitary groups, complementary to the even rank case previously formulated in the author’s thesis.

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Notes

  1. 1.

    It means that V has signature (n − 1,  1) at τ and (n,  0) at other archimedean places, and satisfies \(\mathrm {V}\otimes _E{\mathbf {A}}_E^{\infty }\simeq {\mathbf {V}}^{\infty }\).

  2. 2.

    It is independent of the choice of \((A_{\mu },i_{\mu })\in \mathcal {A}(\mu )\); hence the notation is justified.

  3. 3.

    Here, we use the subscript V to indicate that they are associated to V.

  4. 4.

    Although it is assumed that \( \operatorname {\mathrm {Sh}}(\mathbf {V})\) is proper in [Liu, Section 5], actually the proof of [Liu, Lemma 5.6] works without this assumption.

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Acknowledgements

The author would like to thank the Simons Foundation and the organizers for inviting him to the Simons Symposium Relative Trace Formulas in April 2018 in Germany, from which the work has been benefited. He also thanks the anonymous referee for valuable suggestions to improve the article. The research of the author is partially supported by NSF grant DMS–1702019 and a Sloan Research Fellowship.

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Authors and Affiliations

  1. Department of Mathematics, Yale University, New Haven, CT, USA

    Yifeng Liu

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  1. Yifeng Liu
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Correspondence to Yifeng Liu .

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Editors and Affiliations

  1. Mathematical Institute, University of Bonn, Bonn, Nordrhein-Westfalen, Germany

    Werner Müller

  2. Department of Mathematics, University of California, Berkeley, CA, USA

    Sug Woo Shin

  3. Department of Mathematics, Malott Hall, Cornell University, Ithaca, NY, USA

    Nicolas Templier

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Liu, Y. (2021). Mixed Arithmetic Theta Lifting for Unitary Groups. In: Müller, W., Shin, S.W., Templier, N. (eds) Relative Trace Formulas. Simons Symposia. Springer, Cham. https://doi.org/10.1007/978-3-030-68506-5_10

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