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The global Gan-Gross-Prasad conjecture for unitary groups: the endoscopic case
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  • Published: 27 January 2022

The global Gan-Gross-Prasad conjecture for unitary groups: the endoscopic case

  • Raphaël Beuzart-Plessis1,
  • Pierre-Henri Chaudouard2 &
  • Michał Zydor3 

Publications mathématiques de l'IHÉS volume 135, pages 183–336 (2022)Cite this article

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Abstract

In this paper, we prove the Gan-Gross-Prasad conjecture and the Ichino-Ikeda conjecture for unitary groups \(U_{n}\times U_{n+1}\) in all the endoscopic cases. Our main technical innovation is the computation of the contributions of certain cuspidal data, called ∗-regular, to the Jacquet-Rallis trace formula for linear groups. We offer two different computations of these contributions: one, based on truncation, is expressed in terms of regularized Rankin-Selberg periods of Eisenstein series and Flicker-Rallis intertwining periods introduced by Jacquet-Lapid-Rogawski. The other, built upon Zeta integrals, is expressed in terms of functionals on the Whittaker model. A direct proof of the equality between the two expressions is also given. Finally several useful auxiliary results about the spectral expansion of the Jacquet-Rallis trace formula are provided.

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References

  1. A. Aizenbud and D. Gourevitch, Generalized Harish-Chandra descent, Gelfand pairs, and an Archimedean analog of Jacquet-Rallis’s theorem, Duke Math. J., 149 (2009), 509–567, with an appendix by the authors and Eitan Sayag.

    Article  MathSciNet  MATH  Google Scholar 

  2. J. Arthur, A trace formula for reductive groups I. Terms associated to classes in \(G(\mathbf {Q})\), Duke Math. J., 45 (1978), 911–952.

    Article  MathSciNet  Google Scholar 

  3. J. Arthur, A trace formula for reductive groups II, Compos. Math., 40 (1980), 87–121.

    MATH  Google Scholar 

  4. J. Arthur, The trace formula in invariant form, Ann. Math. (2), 114 (1981), 1–74.

    Article  MathSciNet  MATH  Google Scholar 

  5. J. Arthur, On the inner product of truncated Eisenstein series, Duke Math. J., 49 (1982), 35–70.

    Article  MathSciNet  MATH  Google Scholar 

  6. J. Arthur, A measure on the unipotent variety, Can. J. Math., 37 (1985), 1237–1274.

    Article  MathSciNet  MATH  Google Scholar 

  7. E. Baruch, A proof of Kirillov’s conjecture, Ann. Math. (2), 158 (2003), 207–252.

    Article  MathSciNet  MATH  Google Scholar 

  8. J. Bernstein, \(P\)-invariant distributions on \(\mathrm{GL}(N)\) and the classification of unitary representations of \(\mathrm{GL}(N)\) (non-Archimedean case), in Lie Group Representations, II, Lecture Notes in Math., vol. 1041, College Park, Md., 1982/1983, pp. 50–102, Springer, Berlin, 1984.

    Chapter  Google Scholar 

  9. J. Bernstein, On the support of Plancherel measure, J. Geom. Phys., 5 (1989), 663–710.

    Article  MathSciNet  MATH  Google Scholar 

  10. J. Bernstein and B. Krötz, Smooth Fréchet globalizations of Harish-Chandra modules, Isr. J. Math., 199 (2014), 45–111.

    Article  MATH  Google Scholar 

  11. J. Bernstein and E. Lapid, On the meromorphic continuation of Eisenstein series, 1911.02342, 2019, prepublication.

  12. N. Bourbaki, Éléments de mathématique. Fasc. XXXIII. Variétés différentielles et analytiques. Fascicule de résultats (Paragraphes 1 à 7), Actualités Scientifiques et Industrielles, vol. 1333, Hermann, Paris, 1967.

    MATH  Google Scholar 

  13. R. Beuzart-Plessis, A new proof of Jacquet-Rallis’s fundamental lemma, Duke Math. J., 170 (2021), 2805–2814.

    Article  MathSciNet  MATH  Google Scholar 

  14. R. Beuzart-Plessis, A local trace formula for the Gan-Gross-Prasad conjecture for unitary groups: the Archimedean case, Astérisque, 418 (2020), viii–299, ISBN: 978-2-85629-919-7.

    MathSciNet  MATH  Google Scholar 

  15. R. Beuzart-Plessis, Comparison of local spherical characters and the Ichino-Ikeda conjecture for unitary groups, J. Inst. Math. Jussieu, 20 (2021), 1803–1854.

    Article  MathSciNet  MATH  Google Scholar 

  16. R. Beuzart-Plessis, Archimedean theory and \(\epsilon \)-factors for the Asai Rankin-Selberg integrals, in Relative Trace Formulas, Simons Symposia, Springer, Cham, 2021. https://doi.org/10.1007/978-3-030-68506-5_1.

    Chapter  MATH  Google Scholar 

  17. R. Beuzart-Plessis, Plancherel formula for \(GL_{n}(F)\backslash GL_{n}(E)\) and applications to the Ichino-Ikeda and formal degree conjectures for unitary groups, Invent. Math., 225 (2021), 159–297.

    Article  MathSciNet  MATH  Google Scholar 

  18. R. Beuzart-Plessis, Y. Liu, W. Zhang and X. Zhu, Isolation of cuspidal spectrum, with application to the Gan–Gross–Prasad conjecture, Ann. Math. (2), 194 (2021), 519–584.

    Article  MathSciNet  MATH  Google Scholar 

  19. W. Casselman, Canonical extensions of Harish-Chandra modules to representations of \(G\), Can. J. Math., 41 (1989), 385–438.

    Article  MathSciNet  MATH  Google Scholar 

  20. W. Casselman, Introduction to the Schwartz space of \(\Gamma \backslash G\), Can. J. Math., 41 (1989), 285–320.

    Article  MathSciNet  MATH  Google Scholar 

  21. J. Cogdell, Notes on \(L\)-functions for \(\mathrm{GL}_{n}\), in School on Automorphic Forms on \(\mathrm{GL}(n)\), ICTP Lect. Notes, vol. 21, pp. 75–158, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2008.

    Google Scholar 

  22. P.-H. Chaudouard and M. Zydor, Le transfert singulier pour la formule des traces de Jacquet-Rallis, Compos. Math., 157 (2021), 303–434.

    Article  MathSciNet  MATH  Google Scholar 

  23. J. Dixmier and P. Malliavin, Factorisations de fonctions et de vecteurs indéfiniment différentiables, Bull. Sci. Math. (2), 102 (1978), 307–330.

    MathSciNet  MATH  Google Scholar 

  24. T. Finis and E. Lapid, On the analytic properties of intertwining operators I: global normalizing factors, Bull. Iranian Math. Soc., 43 (2017), 235–277.

    MathSciNet  MATH  Google Scholar 

  25. Y. Flicker, Twisted tensors and Euler products, Bull. Soc. Math. Fr., 116 (1988), 295–313.

    Article  MathSciNet  MATH  Google Scholar 

  26. B. Feigon, E. Lapid and O. Offen, On representations distinguished by unitary groups, Publ. Math. Inst. Hautes Études Sci., 115 (2012), 185–323.

    Article  MathSciNet  MATH  Google Scholar 

  27. J. Franke, Harmonic analysis in weighted \(L_{2}\)-spaces, Ann. Sci. Éc. Norm. Supér. (4), 31 (1998), 181–279.

    Article  MATH  Google Scholar 

  28. J. Franke and J. Schwermer, A decomposition of spaces of automorphic forms, and the Eisenstein cohomology of arithmetic groups, Math. Ann., 311 (1998), 765–790.

    Article  MathSciNet  MATH  Google Scholar 

  29. W. T. Gan, B. Gross and D. Prasad, Symplectic local root numbers, central critical \(L\) values, and restriction problems in the representation theory of classical groups, Astérisque, 346 (2012), 1–109, Sur les conjectures de Gross et Prasad. I.

    MathSciNet  MATH  Google Scholar 

  30. S. Gelbart, H. Jacquet and J. Rogawski, Generic representations for the unitary group in three variables, Isr. J. Math., 126 (2001), 173–237.

    Article  MathSciNet  MATH  Google Scholar 

  31. D. Ginzburg, D. Jiang and S. Rallis, Models for certain residual representations of unitary groups, in Automorphic Forms and \(L\)-Functions I. Global Aspects, Contemp. Math., vol. 488, pp. 125–146, Am. Math. Soc., Providence, 2009.

    Google Scholar 

  32. I. M. Gelfand and D. A. Každan, Representations of the group \(\mathrm{GL}(n,K)\) where \(K\) is a local field, Funkc. Anal. Prilozh., 6 (1972), 73–74.

    MathSciNet  Google Scholar 

  33. H. Grobner and J. Lin, Special values of \(L\)-functions and the refined Gan-Gross-Prasad conjecture, Am. J. Math., 143 (2021), 1–79.

    Article  MathSciNet  MATH  Google Scholar 

  34. D. Goldberg, Some results on reducibility for unitary groups and local Asai \(L\)-functions, J. Reine Angew. Math., 448 (1994), 65–95.

    MathSciNet  MATH  Google Scholar 

  35. A. Grothendieck, Sur certains espaces de fonctions holomorphes. I, J. Reine Angew. Math., 192 (1953), 35–64.

    Article  MathSciNet  MATH  Google Scholar 

  36. A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc., vol. 16, 1955, Chap. 1: 196 pp.; Chap. 2: 140 pp.

    MATH  Google Scholar 

  37. A. Grothendieck, Topological Vector Spaces, Notes on Mathematics and Its Applications, Gordon and Breach, New York, 1973, translated from the French by Orlando Chaljub.

    MATH  Google Scholar 

  38. B. Gross, On the motive of a reductive group, Invent. Math., 130 (1997), 287–313.

    Article  MathSciNet  MATH  Google Scholar 

  39. D. Ginzburg, S. Rallis and D. Soudry, The Descent Map from Automorphic Representations of \(\mathrm{GL}(n)\) to Classical Groups, World Scientific Publishing Co. Pte. Ltd., Hackensack, 2011.

    MATH  Google Scholar 

  40. R. N. Harris, The refined Gross-Prasad conjecture for unitary groups, Int. Math. Res. Not., 2014 (2014), 303–389.

    Article  MathSciNet  MATH  Google Scholar 

  41. A. Ichino and S. Yamana, Periods of automorphic forms: the case of \((\text{GL}_{n+1}\times \text{GL}_{n},\text{GL}_{n})\), Compos. Math., 151 (2015), 665–712.

    Article  MathSciNet  MATH  Google Scholar 

  42. A. Ichino and S. Yamana, Periods of automorphic forms: the case of \((\mathrm{U}_{n+1}\times \mathrm{U}_{n},\mathrm{U}_{n})\), J. Reine Angew. Math., 746 (2019), 1–38.

    Article  MathSciNet  MATH  Google Scholar 

  43. H. Jacquet, Integral representation of Whittaker functions, in Contributions to Automorphic Forms, Geometry, and Number Theory, pp. 373–419, Johns Hopkins Univ. Press, Baltimore, 2004.

    Google Scholar 

  44. H. Jacquet, Archimedean Rankin-Selberg integrals. In automorphic forms and \(L\)-functions II. Local aspects, in Contemp. Math., vol. 489, pp. 57–172, Am. Math. Soc., Providence, 2009.

    Google Scholar 

  45. H. Jacquet, Distinction by the quasi-split unitary group, Isr. J. Math., 178 (2010), 269–324.

    Article  MathSciNet  MATH  Google Scholar 

  46. H. Jacquet, E. Lapid and J. Rogawski, Periods of automorphic forms, J. Am. Math. Soc., 12 (1999), 173–240.

    Article  MathSciNet  MATH  Google Scholar 

  47. H. Jacquet, I. I. Piatetskii-Shapiro and J. A. Shalika, Rankin-Selberg convolutions, Am. J. Math., 105 (1983), 367–464.

    Article  MathSciNet  MATH  Google Scholar 

  48. H. Jacquet and S. Rallis, On the Gross-Prasad conjecture for unitary groups, in On Certain \(L\)-Functions, Clay Math. Proc., vol. 13, pp. 205–264, Am. Math. Soc., Providence, 2011.

    MATH  Google Scholar 

  49. H. Jacquet and J. A. Shalika, On Euler products and the classification of automorphic forms. II, Am. J. Math., 103 (1981), 777–815.

    Article  MathSciNet  MATH  Google Scholar 

  50. H. Jacquet and J. A. Shalika, On Euler products and the classification of automorphic representations. I, Am. J. Math., 103 (1981), 499–558.

    Article  MathSciNet  MATH  Google Scholar 

  51. D. Jiang and L. Zhang, Arthur parameters and cuspidal automorphic modules of classical groups, Ann. Math. (2), 191 (2020), 739–827.

    Article  MathSciNet  MATH  Google Scholar 

  52. A. Kemarsky, A note on the Kirillov model for representations of \(\mathrm{GL}_{n}(\mathbf {C})\), C. R. Math. Acad. Sci. Paris, 353 (2015), 579–582.

    Article  MathSciNet  MATH  Google Scholar 

  53. T. Kaletha, A. Minguez, S.-W. Shin and P.-J. White, Endoscopic classification of representations: inner forms of unitary groups, 1409.3731.

  54. C. Keys and F. Shahidi, Artin \(L\)-functions and normalization of intertwining operators, Ann. Sci. Éc. Norm. Supér. (4), 21 (1988), 67–89.

    Article  MathSciNet  MATH  Google Scholar 

  55. R. Langlands, On the Functional Equations Satisfied by Eisenstein Series, Lecture Notes in Mathematics, vol. 544, Springer, Berlin, 1976.

    MATH  Google Scholar 

  56. E. Lapid, On the fine spectral expansion of Jacquet’s relative trace formula, J. Inst. Math. Jussieu, 5 (2006), 263–308.

    Article  MathSciNet  MATH  Google Scholar 

  57. E. Lapid, A remark on Eisenstein series, in Eisenstein Series and Applications, Progr. Math., vol. 258, pp. 239–249, Birkhäuser, Boston, 2008.

    Chapter  MATH  Google Scholar 

  58. E. Lapid, On Arthur’s asymptotic inner product formula of truncated Eisenstein series, in On Certain \(L\)-Functions, Clay Math. Proc., vol. 13, pp. 309–331, Am. Math. Soc., Providence, 2011.

    MATH  Google Scholar 

  59. E. Lapid, On the Harish-Chandra Schwartz space of \(G(F) \backslash G( \mathbf {A})\), in Automorphic Representations and \(L\)-Functions, Tata Inst. Fundam. Res. Stud. Math., vol. 22, pp. 335–377, Tata Inst. Fund. Res., Mumbai, 2013, with an appendix by Farrell Brumley.

    Google Scholar 

  60. E. Lapid and J. Rogawski, Periods of Eisenstein series: the Galois case, Duke Math. J., 120 (2003), 153–226.

    Article  MathSciNet  MATH  Google Scholar 

  61. J.-P. Labesse and J.-L. Waldspurger, La formule des traces tordue d’après le Friday Morning Seminar, CRM Monograph Series, vol. 31, Am. Math. Soc., Providence, 2013, with a foreword by Robert Langlands [dual English/French text].

    MATH  Google Scholar 

  62. C. P. Mok, Endoscopic Classification of Representations of Quasi-Split Unitary Groups, Mem. Amer. Math. Soc., vol. 235(1108), vi+248, 2015.

    MATH  Google Scholar 

  63. W. Müller and B. Speh, Absolute convergence of the spectral side of the Arthur trace formula for \(\mathrm{GL}_{n}\), Geom. Funct. Anal., 14 (2004), 58–93, with an appendix by E. M. Lapid.

    Article  MathSciNet  MATH  Google Scholar 

  64. C. Mœglin and J.-L. Waldspurger, Le spectre résiduel de \(\mathrm{GL}(n)\), Ann. Sci. Éc. Norm. Supér. (4), 22 (1989), 605–674.

    Article  MATH  Google Scholar 

  65. C. Mœglin and J.-L. Waldspurger, Décomposition spectrale et séries d’Eisenstein, Progress in Mathematics, vol. 113, Birkhäuser, Basel, 1994, Une paraphrase de l’Écriture. [A paraphrase of Scripture].

    MATH  Google Scholar 

  66. W. Müller, On the singularities of residual intertwining operators, Geom. Funct. Anal., 10 (2000), 1118–1170.

    Article  MathSciNet  MATH  Google Scholar 

  67. D. Ramakrishnan, A Theorem on GL(n) à la Tchebotarev, arXiv e-prints, 1806.08429, Jun 2018.

  68. F. Shahidi, On certain \(L\)-functions, Am. J. Math., 103 (1981), 297–355.

    Article  MATH  Google Scholar 

  69. F. Shahidi, A proof of Langlands’ conjecture on Plancherel measures; complementary series for \(p\)-adic groups, Ann. Math. (2), 132 (1990), 273–330.

    Article  MathSciNet  MATH  Google Scholar 

  70. F. Trèves, Topological Vector Spaces, Distributions and Kernels, Academic Press, New York, 1967.

    MATH  Google Scholar 

  71. N. Wallach, Real Reductive Groups. II, Pure and Applied Mathematics, vol. 132, Academic Press, Boston, 1992.

    MATH  Google Scholar 

  72. A. Weil, Adeles and Algebraic Groups, Progress in Mathematics, vol. 23, Birkhäuser, Boston, 1982, with appendices by M. Demazure and Takashi Ono.

    Book  MATH  Google Scholar 

  73. H. Xue, On the global Gan-Gross-Prasad conjecture for unitary groups: approximating smooth transfer of Jacquet-Rallis, J. Reine Angew. Math., 756 (2019), 65–100.

    Article  MathSciNet  MATH  Google Scholar 

  74. Z. Yun, The fundamental lemma of Jacquet and Rallis, Duke Math. J., 156 (2011), 167–227, with an appendix by Julia Gordon.

    MathSciNet  MATH  Google Scholar 

  75. W. Zhang, Automorphic period and the central value of Rankin-Selberg \(L\)-function, J. Am. Math. Soc., 27 (2014), 541–612.

    Article  MathSciNet  MATH  Google Scholar 

  76. W. Zhang, Fourier transform and the global Gan-Gross-Prasad conjecture for unitary groups, Ann. Math. (2), 180 (2014), 971–1049.

    Article  MathSciNet  MATH  Google Scholar 

  77. M. Zydor, La variante infinitésimale de la formule des traces de Jacquet-Rallis pour les groupes unitaires, Can. J. Math., 68 (2016), 1382–1435.

    Article  MathSciNet  MATH  Google Scholar 

  78. M. Zydor, La variante infinitésimale de la formule des traces de Jacquet-Rallis pour les groupes linéaires, J. Inst. Math. Jussieu, 17 (2018), 735–783.

    Article  MathSciNet  MATH  Google Scholar 

  79. M. Zydor, Les formules des traces relatives de Jacquet–Rallis grossières, J. Reine Angew. Math., 762 (2020), 195–259.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. CNRS, Centrale Marseille, I2M, Aix Marseille Univ, Marseille, France

    Raphaël Beuzart-Plessis

  2. Université de Paris et Sorbonne Université, CNRS, IMJ-PRG, 75013, Paris, France

    Pierre-Henri Chaudouard

  3. University of Michigan, Ann Arbor, MI, USA

    Michał Zydor

Authors
  1. Raphaël Beuzart-Plessis
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  2. Pierre-Henri Chaudouard
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  3. Michał Zydor
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Correspondence to Pierre-Henri Chaudouard.

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Beuzart-Plessis, R., Chaudouard, PH. & Zydor, M. The global Gan-Gross-Prasad conjecture for unitary groups: the endoscopic case. Publ.math.IHES 135, 183–336 (2022). https://doi.org/10.1007/s10240-021-00129-1

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  • Received: 17 December 2021

  • Accepted: 20 December 2021

  • Published: 27 January 2022

  • Issue Date: June 2022

  • DOI: https://doi.org/10.1007/s10240-021-00129-1

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