On Relative Trace Formulae: the Case of Jacquet-Rallis

Abstract

We give an account of recent works on Jacquet-Rallis’ approach to the Gan-Gross-Prasad conjecture for unitary groups. We report on the present state of the Jacquet-Rallis relative trace formulae and on some current applications of it. We give also a precise computation of the constant that appears in the statement “Fourier transform and transfer commute up to a constant”.

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Notes

  1. 1.

    Since it may be a source of confusion, we emphasize that it is f(a, t) in the expression below and not f(t, a).

  2. 2.

    For a better result see the very recent preprint [10].

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Acknowledgements

I would like to thank the organizers of the VIASM Annual Meeting 2017 for the invitation to give a lecture and to offer me the opportunity to write this article for the proceedings. I would also like to thank them and especially Ngô Bao Châu for the wonderful stay in Vietnam.

During the preparation of this article, I received partial support from Institut Universitaire de France and Agence Nationale pour la Recherche (projects Ferplay ANR-13-BS01-0012 and Vargen ANR-13-BS01-0001-01).

I also thank Michał, Zydor for numerous discussions on the topics of the present article.

Finally, I thank Hang Xue for helpful correspondence. I thank the anonymous referee for her/his careful proofreading.

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Lecture at the Annual Meeting 2017 of the Vietnam Institute for Advanced Study in Mathematics

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Chaudouard, P. On Relative Trace Formulae: the Case of Jacquet-Rallis. Acta Math Vietnam 44, 391–430 (2019). https://doi.org/10.1007/s40306-018-00312-3

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Keywords

  • Langlands program
  • Relative trace formula
  • Orbital integrals
  • Gan-Gross-Prasad conjectures

Mathematics Subject Classification (2010)

  • 11F70
  • 22E50
  • 22E55
  • 11F66
  • 11R39