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Bessel identities in the waldspurger correspondence over the real numbers

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Abstract

We prove certain identities between Bessel functions attached to irreducible unitary representations ofPGL 2(R) and Bessel functions attached to irreducible unitary representations of the double cover ofSL 2(R). These identities give a correspondence between such representations which turns out to be the Waldspurger correspondence. In the process we prove several regularity theorems for Bessel distributions which appear in the relative trace formula. In the heart of the proof lies a classical result of Weber and Hardy on a Fourier transform of classical Bessel functions. This paper constitutes the local (real) spectral theory of the relative trace formula for the Waldspurger correspondence for which the global part was developed by Jacquet.

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Authors and Affiliations

  1. Department of Mathematics, Technion, 32000, Haifa, Israel

    Ehud Moshe Baruch

  2. Department of Mathematics, Rutgers University, 07102, Newark, NJ, USA

    Zhengyu Mao

Authors
  1. Ehud Moshe Baruch
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  2. Zhengyu Mao
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Corresponding author

Correspondence to Ehud Moshe Baruch.

Additional information

Research of first author was partially supported by NSF grant DMS-0070762.

Research of second author was partially supported by NSF grant DMS-9729992 and DMS 9971003.

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Baruch, E.M., Mao, Z. Bessel identities in the waldspurger correspondence over the real numbers. Isr. J. Math. 145, 1–81 (2005). https://doi.org/10.1007/BF02786684

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  • DOI: https://doi.org/10.1007/BF02786684

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