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The Shimura–Waldspurger Correspondence for Mp(2n)

  • Wee Teck Gan
  • Wen-Wei Li
Conference paper
Part of the Simons Symposia book series (SISY)

Abstract

We describe some recent developments and formulate some conjectures in the genuine representation theory and the study of automorphic forms of the metaplectic group Mp(2n), from the point of view of the theta correspondence as well as from the point of view of the theory of endoscopy and the trace formula.

Keywords

Shimura-Waldspurger correspondence Theta lifting Transfer Local character identity Local intertwining relation 

Notes

Acknowledgements

We thank the Simons Foundation for its generous travel and local support during the duration of the Simons Symposium. We are also grateful to Caihua Luo for his comments on an earlier draft.

References

  1. 1.
    J. Adams, Lifting of characters on orthogonal and metaplectic groups, Duke Mathematical Journal 92 (1998), no. 1, 129–178.MathSciNetCrossRefGoogle Scholar
  2. 2.
    J. Adams and D. Barbasch, Reductive dual pair correspondence for complex groups, J. Funct. Anal. 132 (1995), no. 1, 1–42.MathSciNetCrossRefGoogle Scholar
  3. 3.
    J. Adams and D. Barbasch, Genuine representations of the metaplectic group, Compos. Math. 113 (1998), no. 1, 23–66.MathSciNetCrossRefGoogle Scholar
  4. 4.
    J. Arthur, The endoscopic classification of representations: orthogonal and symplectic groups, American Mathematical Society Colloquium Publications 61, American Mathematical Society, Providence, RI, 2013.CrossRefGoogle Scholar
  5. 5.
    P. Deligne and J.-L. Brylinski, Central extensions of reductive groups by K 2, Publications Mathématiques de l’IHÉS, 94 (2001), p. 5–85.MathSciNetCrossRefGoogle Scholar
  6. 6.
    D. Bump, S. Friedberg and J. Hoffstein, p-adic Whittaker functions on the metaplectic group, Duke Math. J. 63 (1991), no. 2, 379–397.MathSciNetCrossRefGoogle Scholar
  7. 7.
    S. Friedberg and J. Hoffstein, Nonvanishing theorems for automorphic L-functions on GL(2), Ann. of Math. (2) 142 (1995), no. 2, 385–423.MathSciNetCrossRefGoogle Scholar
  8. 8.
    W. T. Gan, The Shimura correspondence à la Waldspurger, http://www.math.nus.edu.sg/~matgwt/postech.pdf.
  9. 9.
    W. T. Gan, B. H. Gross, and D. Prasad, Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups, Sur les conjectures de Gross et Prasad. I, Astérisque No. 346 (2012), 1–109.Google Scholar
  10. 10.
    W. T. Gan and A. Ichino, The Shimura-Waldspurger correspondence for Mp2n, preprint, available at https://arxiv.org/pdf/1705.10106.pdf.
  11. 11.
    W. T. Gan and G. Savin, Representations of metaplectic groups I: epsilon dichotomy and local Langlands correspondence, Compos. Math. 148 (2012), no. 6, 1655–1694.MathSciNetCrossRefGoogle Scholar
  12. 12.
    T. Kaletha, Epipelagic L-packets and rectifying characters, Invent. Math. 202 (2015), 1–89.MathSciNetCrossRefGoogle Scholar
  13. 13.
    T. Kaletha, A. Mínguez, S. W. Shin, and P.-J. White. Endoscopic classification of representations: inner forms of unitary groups, arXiv:1409.3731.Google Scholar
  14. 14.
    R. P. Langlands, On the notion of an automorphic representation, Proc. Sympos. Pure Math. 33, Automorphic forms, representations and L-functions, Part 1, pp. 203–207, Amer. Math. Soc., Providence, R.I., 1979.Google Scholar
  15. 15.
    J.-S. Li, Singular unitary representations of classical groups, Invent. Math. 97 (1989), no. 2, 237–255.MathSciNetCrossRefGoogle Scholar
  16. 16.
    J.-S. Li, On the classification of irreducible low rank unitary representations of classical groups, Compos. Math. 71 (1989), no. 1, 29–48.MathSciNetzbMATHGoogle Scholar
  17. 17.
    J.-S. Li, Automorphic forms with degenerate Fourier coefficients, Amer. J. Math. 119 (1997), no. 3, 523–578.MathSciNetCrossRefGoogle Scholar
  18. 18.
    W.-W. Li, Transfert d’intégrales orbitales pour le groupe métaplectique, Compos. Math. 147 (2011), no. 2, 524–590.MathSciNetCrossRefGoogle Scholar
  19. 19.
    W.-W. Li, Le lemme fondamental pondéré pour le groupe métaplectique, Canad. J. Math. 64 (2012), no. 3, 497–543.MathSciNetCrossRefGoogle Scholar
  20. 20.
    W.-W. Li, La formule des traces stable pour le groupe métaplectique: les termes elliptiques, Invent. Math. 202 (2015), no. 2, 743–838.MathSciNetCrossRefGoogle Scholar
  21. 21.
    W.-W. Li, Spectral transfer for metaplectic groups. I. Local character relations, to appear in the Journal of the Inst. of Math. Jussieu, arXiv:1409.6106.Google Scholar
  22. 22.
    C.H. Luo, Spherical fundamental lemma for metaplectic groups, Canad. J. Math. 70 (2018), no. 4, 898–924.MathSciNetCrossRefGoogle Scholar
  23. 23.
    C.H. Luo, Endoscopic character identities for metaplectic groups, preprint.Google Scholar
  24. 24.
    C. Moeglin and J.-L. Waldspurger, Spectral decomposition and Eisenstein series, Cambridge Tracts in Mathematics 113, Cambridge University Press, 1995.CrossRefGoogle Scholar
  25. 25.
    C. Mœglin, Paquets d’Arthur discrets pour un groupe classique p-adique, Automorphic forms and L-functions II. Local aspects, 179–257, Contemp. Math. 489, Amer. Math. Soc., Providence, RI, 2009.Google Scholar
  26. 26.
    C. Mœglin, Multiplicité 1 dans les paquets d’Arthur aux places p-adiques, On certain L-functions, 333–374, Clay Math. Proc. 13, Amer. Math. Soc., Providence, RI, 2011.Google Scholar
  27. 27.
    S. Niwa, Modular forms of half integral weight and the integral of certain theta-functions, Nagoya Math. J. 56 (1975), 147–161.MathSciNetCrossRefGoogle Scholar
  28. 28.
    D. Renard, Endoscopy for \(\mathrm {Mp}(2n, \mathbb {R})\), American Journal of Mathematics 121 (1999), no. 6, 1215–1244.MathSciNetCrossRefGoogle Scholar
  29. 29.
    G. Shimura, On modular forms of half integral weight, Ann. of Math. (2) 97 (1973), 440–481.MathSciNetCrossRefGoogle Scholar
  30. 30.
    T. Shintani, On construction of holomorphic cusp forms of half integral weight, Nagoya Math. J. 58 (1975), 83–126.MathSciNetCrossRefGoogle Scholar
  31. 31.
    T. Springer, Linear algebraic groups, Second edition, Prog. Math. 9 (1998). Birkhäuser.Google Scholar
  32. 32.
    D. Szpruch, Some irreducibility theorems of parabolic induction on the metaplectic group via the Langlands-Shahidi method, Israel J. Math. 195 (2013), no. 2, 897–971.MathSciNetCrossRefGoogle Scholar
  33. 33.
    J.-L. Waldspurger, Correspondance de Shimura, J. Math. Pures Appl. (9) 59 (1980), no. 1, 1–132.Google Scholar
  34. 34.
    J.-L. Waldspurger, Correspondances de Shimura et quaternions, Forum Math. 3 (1991), no. 3, 219–307.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsNational University of SingaporeSingaporeSingapore
  2. 2.Beijing International Center for Mathematical ResearchBeijingPeople’s Republic of China
  3. 3.School of Mathematical SciencesPeking UniversityBeijingPeople’s Republic of China

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