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Recent progress on the Gross–Prasad conjecture

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We report on recent progress on both the local and global Gross–Prasad conjectures for unitary groups.

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References

  1. Aizenbud, A., Gourevitch, D.: Generalized Harish-Chandra descent, Gelfand pairs, and an Archimedean analog of Jacquet–Rallis’s theorem. Duke Math. J. 149(3), 509–567 (2009). With an appendix by the authors and Eitan Sayag

    Article  MATH  MathSciNet  Google Scholar 

  2. Aizenbud, A., Gourevitch, D., Rallis, S., Schiffmann, G.: Multiplicity one theorems. Ann. Math. 172(2), 1407–1434 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  3. Arthur, J.: The endoscopic classification of representations: orthogonal and symplectic groups. To appear as a Colloquium Publication of the American Mathematical Society (2013)

  4. Beuzart-Plessis, R.: La conjecture locale de Gross–Prasad pour les représentations tempérées des groupes unitaires. Preprint (2012)

  5. Deligne, P.: Les constantes des équations fonctionnelles des fonctions L. In: Modular Functions on One Variable, II. Lecture Notes in Mathematics, vol. 349, pp. 501–597. Springer, Berlin (1973)

    Chapter  Google Scholar 

  6. Gan, W.T.: Representations of metaplectic groups. In: Fifth International Congress of Chinese Mathematicians. Parts 1, 2. AMS/IP Stud. Adv. Math., vol. 51, pp. 155–170. Am. Math. Soc., Providence (2012)

    Google Scholar 

  7. Gan, W.T., Ichino, A.: The Gross–Prasad conjecture and local theta correspondence. Preprint

  8. Gan, W.T., Gross, B.H., Prasad, D.: Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups. Astérisque 346, 1–110 (2012)

    Google Scholar 

  9. Gan, W.T., Gross, B.H., Prasad, D.: Restrictions of representations of classical groups: examples. Astérisque 346, 111–170 (2012)

    Google Scholar 

  10. Ginzburg, D., Jiang, D., Rallis, S.: On the nonvanishing of the central value of the Rankin–Selberg L-functions. J. Am. Math. Soc. 17(3), 679–722 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  11. Ginzburg, D., Jiang, D., Rallis, S.: On the nonvanishing of the central value of the Rankin–Selberg L-functions, II. In: Automorphic Representations, L-functions and Applications: Progress and Prospects. Ohio State Univ. Math. Res. Inst. Publ., vol. 11, pp. 157–191. de Gruyter, Berlin (2005)

    Google Scholar 

  12. Ginzburg, D., Jiang, D., Rallis, S.: Models for certain residual representations of unitary groups. Automorphic forms and L-functions I. Global aspects. In: Contemp. Math., vol. 488, pp. 125–146. Am. Math. Soc., Providence (2009)

    Google Scholar 

  13. Gross, B., Prasad, D.: On the decomposition of a representation of SO n when restricted to SO n−1. Can. J. Math. 44(5), 974–1002 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  14. Gross, B., Prasad, D.: On irreducible representations of SO2n+1×SO2m . Can. J. Math. 46(5), 930–950 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  15. Harris, N.: The refined Gross–Prasad conjecture for unitary groups. Int. Math. Res. Not. 2014, 303–389 (2014). doi:10.1093/imrn/rns219

    Google Scholar 

  16. Harris, M., Taylor, R.: The geometry and cohomology of some simple Shimura varieties. Annals of Mathematics Studies, vol. 151. Princeton University Press, Princeton (2001). With an appendix by Vladimir G. Berkovich

    MATH  Google Scholar 

  17. Henniart, G.: Caractérisation de la correspondance de Langlands locale par les facteurs? De paires. Invent. Math. 113(2), 339–350 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  18. Henniart, G.: Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique. Invent. Math. 139(2), 439–455 (2010)

    Article  MathSciNet  Google Scholar 

  19. 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)

    Article  MATH  MathSciNet  Google Scholar 

  20. Jacquet, H., Rallis, S.: On the Gross–Prasad conjecture for unitary groups. On certain L-functions. In: Clay Math. Proc., vol. 13, pp. 205–264. Am. Math. Soc., Providence (2011)

    Google Scholar 

  21. Kaletha, T.: Rigid inner forms of real and p-adic groups. Preprint, available at: arXiv:1304.3292 [math.RT]

  22. Kottwitz, R., Shelstad, D.: Foundations of twisted endoscopy. Astérisque 255 (1999)

  23. Liu, Y.F.: Relative trace formulae towards Bessel and Fourier–Jacobi periods of unitary groups. To appear in Manuscr. Math.

  24. Moeglin, C., Waldspurger, J.-L.: La conjecture locale de Gross–Prasad pour les groupes spéciaux orthogonaux: le cas général. Astérisque 347 (2012)

  25. Mok, C.P.: Endoscopic classification of representations of quasi-split unitary groups. To appear in Memoirs of AMS

  26. Sun, B.: Multiplicity one theorems for Fourier–Jacobi models. Am. J. Math. 134(6), 1655–1678 (2012)

    Article  MATH  Google Scholar 

  27. Sun, B., Zhu, C.-B.: Multiplicity one theorems: the Archimedean case. Ann. Math. 175(1), 23–44 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  28. Vogan, D.: The local Langlands conjecture. Contemp. Math. 145, 305–379 (1993)

    Article  MathSciNet  Google Scholar 

  29. Waldspurger, J.-L.: Une formule intégrale reliée à la conjecture locale de Gross–Prasad. Compos. Math. 146, 1180–1290 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  30. Waldspurger, J.-L.: Une formule intégrale reliée à la conjecture locale de Gross–Prasad, 2ème partie: Extension aux représentations tempérées. Astérisque 346, 171–311 (2012)

    Google Scholar 

  31. Waldspurger, J.-L.: Calcul d’une valeur d’un facteur epsilon par une formule intégrale. Astérisque 347 (2012)

  32. Waldspurger, J.-L.: La conjecture locale de Gross–Prasad pour les représentations tempérées des groupes spéciaux orthogonaux. Astérisque 347 (2012)

  33. Waldspurger, J.-L.: Une variante d’un résultat de Aizenbud, Gourevitch, Rallis et Schiffmann. Astérisque 346, 313–318 (2012)

    Google Scholar 

  34. Xue, H.: The Gan–Gross–Prasad conjecture for U(n)×U(n). Preprint (2012). Available at: http://math.columbia.edu/~xuehang/ggp.pdf

  35. Xue, H.: Fourier–Jacobi periods and the central value of Rankin–Selberg L-function. Preprint (2013)

  36. Yuan, X., Zhang, S.W., Zhang, W.: The Gross–Zagier formula on Shimura curves. Ann. Math. Stud. 184, viii+272 (2013)

    Google Scholar 

  37. Yun, Z.W.: The fundamental lemma of Jacquet and Rallis. Math. J. 156(2), 167–227 (2011). With an appendix by Julia Gordon, Duke

    MATH  MathSciNet  Google Scholar 

  38. Zhang, W.: Fourier transform and the global Gan–Gross–Prasad conjecture for unitary groups. To appear in Annals of Math. Available at: http://www.math.columbia.edu/~wzhang/math/online/transfer.pdf

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

    Article  Google Scholar 

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Acknowledgements

This paper is a written account of my lecture delivered at the annual meeting of the Vietnam Institute for Advanced Study in Mathematics held on July 20–21, 2013. I thank Bao Chau Ngo for his invitation to deliver a lecture and the local organisers, especially Professor Ho Hai Phung, for their warm hospitality and travel support.

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Correspondence to Wee Teck Gan.

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Gan, W.T. Recent progress on the Gross–Prasad conjecture. Acta Math Vietnam. 39, 11–33 (2014). https://doi.org/10.1007/s40306-014-0047-2

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