Acta Mathematica Vietnamica

, Volume 44, Issue 2, pp 391–430 | Cite as

On Relative Trace Formulae: the Case of Jacquet-Rallis

  • Pierre-Henri ChaudouardEmail author


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”.


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

Mathematics Subject Classification (2010)

11F70 22E50 22E55 11F66 11R39 



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.


  1. 1.
    Aizenbud, A., Gourevitch, D., Rallis, S., Schiffmann, G.: Multiplicity one theorems. Ann. of Math. (2) 172(2), 1407–1434 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aizenbud, A.: A partial analog of the integrability theorem for distributions on p-adic spaces and applications. Israel J. Math. 193(1), 233–262 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Arthur, J.: A trace formula for reductive groups I. Terms associated to classes in \(G(\mathbb {Q})\). Duke Math. J. 45, 911–952 (1978)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Arthur, J.: An introduction to the trace formula. In: Harmonic Analysis, the Trace Formula, and Shimura Varieties, vol. 4. Clay Math. Proc., 1–263. Amer. Math. Soc., Providence, RI (2005)Google Scholar
  5. 5.
    Arthur, J.: The endoscopic classification of representations. American Mathematical Society Colloquium Publications, vol. 61. American Mathematical Society, Providence (2013). Orthogonal and symplectic groupsGoogle Scholar
  6. 6.
    Beuzart-Plessis, R.: Cours Peccot (2017)Google Scholar
  7. 7.
    Beuzart-Plessis, R.: La conjecture locale de Gross-Prasad pour les représentations tempérées des groupes unitaires. Preprint arXiv:1205.2987v2(2012)
  8. 8.
    Beuzart-Plessis, R.: A local trace formula for the Gan-Gross-Prasad conjecture for unitary groups: the archimedean case. ArXiv e-prints (2015)Google Scholar
  9. 9.
    Beuzart-Plessis, R.: Comparison of local spherical characters and the Ichino-Ikeda conjecture for unitary groups. ArXiv e-prints (2016)Google Scholar
  10. 10.
    Beuzart-Plessis, R.: Plancherel formula for G L n(F)∖G L n(E) and applications to the Ichino-Ikeda and formal degree conjectures for unitary groups. Preprint (2018)Google Scholar
  11. 11.
    Bushnell, C., Henniart, G.: The local Langlands conjecture for GL(2) Grundlehren Der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 335. Springer, Berlin (2006)Google Scholar
  12. 12.
    Chaudouard, P.-H.: La formule des traces pour les algèbres de Lie. Math. Ann. 322(2), 347–382 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Chaudouard, P.-H., Zydor, M.: Le transfert singulier pour la formule des traces de Jacquet-Rallis. Preprint,
  14. 14.
    Flicker, Y.: Twisted tensors and Euler products. Bull. Soc. Math. France 116 (3), 295–313 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gan, W.T., Gross, B., Prasad, D.: Symplectic local root numbers, central critical L values, and restriction problems in the representation theory of classical groups. Astérisque 346(1), 1–109 (2012). Sur les conjectures de Gross et Prasad. IMathSciNetzbMATHGoogle Scholar
  16. 16.
    Harris, R.N.: The refined Gross-Prasad conjecture for unitary groups. Int. Math. Res. Not. IMRN 2, 303–389 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Harris, M., Labesse, J.-P.: Conditional base change for unitary groups. Asian J. Math. 8(4), 653–683 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ichino, A., Ikeda, T.: On the periods of automorphic forms on special orthogonal groups and the Gross-Prasad conjecture. Geom. Funct. Anal. 19(5), 1378–1425 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Jacquet, H.: Automorphic spectrum of symmetric spaces. In: Representation Theory and Automorphic Forms (Edinburgh, 1996). Proc. Sympos. Pure Math., vol. 61. Am. Math. Soc., 443–455, Providence, RI (1997)Google Scholar
  20. 20.
    Jacquet, H., Piatetskii-Shapiro, I.I., Shalika, J. A.: Rankin-selberg convolutions. Am. J. Math. 105(2), 367–464 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Jacquet, H., Rallis, S.: On the Gross-Prasad conjecture for unitary groups. In: On Certain L-Functions. Clay Math. Proc., vol. 13, 205–265, Am. Math. Soc. (2011)Google Scholar
  22. 22.
    Kaletha, T., Minguez, A., Shin, S. W., White, P.-J.: Endoscopic classification of representations: inner forms of unitary groups. ArXiv e-prints (2014)Google Scholar
  23. 23.
    Kottwitz, R.: Transfer factors for Lie algebras. Represent. Theory 3, 127–138 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Langlands, R.: On the functional equations satisfied by Eisenstein series lecture notes in mathematics, vol. 544. Springer, Berlin (1976)CrossRefGoogle Scholar
  25. 25.
    Labesse, J.-P., Waldspurger, J.-L.: La Formule Des Traces Tordue D’après Le Friday Morning Seminar. CRM Monograph Series, vol. 31. American Mathematical Society, Providence (2013). With a foreword by Robert Langlands [dual English/French text]CrossRefzbMATHGoogle Scholar
  26. 26.
    Mok, C.P.: Endoscopic classification of representations of quasi-split unitary groups. Mem. Am. Math. Soc. 235, vi+ 248 (2015)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Mœglin, C.: Décomposition Spectrale et Séries d’Eisenstein. Progress in Mathematics, vol. 113. Birkhäuser Verlag, Basel (1994). Une paraphrase de l’ÉcritureGoogle Scholar
  28. 28.
    Ramakrishnan, D.: A mild Tchebotarev theorem for GL(n). J. Number Theory 146, 519–533 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Rallis, S., Schiffman, G.: Multiplicity one conjectures prépublication. arXiv:0705.21268v1 (2008)
  30. 30.
    Ramakrishnan, D., Valenza, R.: Fourier analysis on number fields graduate texts in mathematics, vol. 186. Springer, New York (1999)CrossRefGoogle Scholar
  31. 31.
    Scharlau, W.: Quadratic and Hermitian Forms Grundlehren Der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 270. Springer, Berlin (1985)Google Scholar
  32. 32.
    Sakellaridis, Y., Venkatesh, A.: Periods and harmonic analysis on spherical varieties. Astérisque 396, viii+ 360 (2017)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Sun, B., Zhu, C.-B.: Multiplicity one theorems: the Archimedean case. Ann. of Math. (2) 175(1), 23–44 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Xue, H.: On the global Gan-Gross-Prasad conjecture for unitary groups: approximating smooth transfer of Jacquet-Rallis. J. Reine Angew. Math. (2015)Google Scholar
  35. 35.
    Yun, Z.: The fundamental lemma of Jacquet and Rallis. Duke Math. J. 156(2), 167–227 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Zhang, W.: On the smooth transfer conjecture of Jacquet-Rallis for n = 3. Ramanujan J. 29(1–3), 225–256 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Zhang, W.: Automorphic period and the central value of Rankin-Selberg L-function. J. Am. Math. Soc. 27, 541–612 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Zhang, W.: Fourier transform and the global Gan-Gross-Prasad conjecture for unitary groups. Ann. of Math. (2) 180(3), 971–1049 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Zydor, M.: Les formules des traces relatives de Jacquet-Rallis grossières. ArXiv e-prints (2015)Google Scholar
  40. 40.
    Zydor, M.: La variante infinitésimale de la formule des traces de Jacquet-Rallis pour les groupes unitaires. Canad. J. Math. 68(6), 1382–1435 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Zydor, M.: La variante infinitésimale de la formule des traces de Jacquet-Rallis pour les groupes linéaires. J. Inst. Math. Jussieu 17(4), 735–783 (2018)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Institut de Mathématiques de Jussieu-Paris Rive GaucheUniversité Paris Diderot (Paris 7) et Institut Universitaire de France, UMR 7586, Bâtiment Sophie GermainParis Cedex 13France

Personalised recommendations