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Archimedean Theory and 𝜖-Factors for the Asai Rankin-Selberg Integrals

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Relative Trace Formulas

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

Abstract

In this paper, we partially complete the local Rankin-Selberg theory of Asai L-functions and 𝜖-factors as introduced by Flicker and Kable. In particular, we establish the relevant local functional equation at Archimedean places and prove the equality between Rankin-Selberg’s and Langlands-Shahidi’s 𝜖-factors at every place. Our proofs work uniformly for any characteristic zero local field and use as only input the global functional equation and a globalization result for a dense subset of tempered representations that we infer from work of Finis-Lapid-Müller. The results of this paper are used in (R. Beuzart-Plessis: Plancherel formula for \( \operatorname {\mathrm {GL}}_n(F)\backslash \operatorname {\mathrm {GL}}_n(E)\) and applications to the Ichino-Ikeda and formal degree conjectures for unitary groups, Preprint 2018) to establish an explicit Plancherel decomposition for \( \operatorname {\mathrm {GL}}_n(F)\backslash \operatorname {\mathrm {GL}}_n(E)\), E∕F a quadratic extension of local fields, with applications to the Ichino-Ikeda and formal degree conjecture for unitary groups.

The project leading to this publication has received funding from Excellence Initiative of Aix-Marseille University-A*MIDEX, a French “Investissements d’Avenir” programme.

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Author information

Authors and Affiliations

  1. Aix-Marseille University, CNRS, Marseille, France

    Raphaël Beuzart-Plessis

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  1. Raphaël Beuzart-Plessis
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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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Beuzart-Plessis, R. (2021). Archimedean Theory and 𝜖-Factors for the Asai Rankin-Selberg Integrals. 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_1

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