Advertisement

Fourier Coefficients and Cuspidal Spectrum for Symplectic Groups

  • Dihua Jiang
  • Baiying Liu
Conference paper
Part of the Simons Symposia book series (SISY)

Abstract

J. Arthur (The endoscopic classification of representations: orthogonal and Symplectic groups. Colloquium Publication, vol 61. American Mathematical Society, 2013) classifies the automorphic discrete spectrum of symplectic groups up to global Arthur packets. We continue with our investigation of Fourier coefficients and their implication to the structure of the cuspidal spectrum for symplectic groups (Jiang, Automorphic integral transforms for classical groups I: endoscopy correspondences. In: Automorphic forms and related geometry: assessing the legacy of I.I. Piatetski-Shapiro. Contemp. Math., vol 614, pp 179–242. AMS, 2014; Jiang and Liu, Fourier coefficients for automorphic forms on quasisplit classical groups. In: Advances in the theory of automorphic forms and their L-functions. Contemp. Math., vol 664, pp 187–208. AMS, 2016). As a result, we obtain certain characterization and construction of small cuspidal automorphic representations and gain a better understanding of global Arthur packets and of the structure of local unramified components of the cuspidal spectrum, which has impacts to the generalized Ramanujan problem as posted by P. Sarnak (Notes on the generalized Ramanujan conjectures. In: Harmonic analysis, the trace formula, and Shimura varieties. Clay Math. Proc., vol 4, pp 659–685. Amer. Math. Soc., Providence, RI, 2005).

Keywords

Arthur parameters and Arthur packets Automorphic discrete spectrum of classical groups Fourier coefficients and small cuspidal automorphic forms 

Notes

Acknowledgements

The research of the first named author is supported in part by the NSF Grants DMS-1301567 and DMS-1600685, and that of the second named author is supported in part by NSF Grants DMS-1620329, DMS-1702218, and start-up funds from Purdue University.

This material is based upon work supported by the National Science Foundation under agreement No. DMS-1128155. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

References

  1. 1.
    P. Achar, An order-reversing duality map for conjugacy classes in Lusztig’s canonical quotient. Transform. Groups 8 (2003), no. 2, 107–145.MathSciNetCrossRefGoogle Scholar
  2. 2.
    J. Arthur, The endoscopic classification of representations: Orthogonal and Symplectic groups. Colloquium Publication Vol. 61, 2013, American Mathematical Society.Google Scholar
  3. 3.
    D. Barbasch and D. Vogan, Unipotent representations of complex semisimple groups. Ann. of Math. (2) 121 (1985), no. 1, 41–110.MathSciNetCrossRefGoogle Scholar
  4. 4.
    V. Blomer and F. Brumley, On the Ramanujan conjecture over number fields. Ann. of Math. (2) 174 (2011), no. 1, 581–605.MathSciNetCrossRefGoogle Scholar
  5. 5.
    D. Collingwood and W. McGovern, Nilpotent orbits in semisimple Lie algebras. Van Nostrand Reinhold Mathematics Series. Van Nostrand Reinhold Co., New York, 1993. xiv+186 pp.Google Scholar
  6. 6.
    W. Duke and Ö. Imamoglu, A converse theorem and the Saito-Kurokawa lift. Internat. Math. Res. Notices 7 (1996), 347–355.MathSciNetCrossRefGoogle Scholar
  7. 7.
    D. Ginzburg, Certain conjectures relating unipotent orbits to automorphic representations. Israel J. Math. 151 (2006), 323–355.MathSciNetCrossRefGoogle Scholar
  8. 8.
    D. Ginzburg, S. Rallis and D. Soudry, On Fourier coefficients of automorphic forms of symplectic groups. Manuscripta Math. 111 (2003), no. 1, 1–16.MathSciNetCrossRefGoogle Scholar
  9. 9.
    D. Ginzburg, S. Rallis and D. Soudry, Contruction of CAP representations for symplectic groups using the descent method. Automorphic representations, L-functions and applications: progress and prospects, 193–224, Ohio State Univ. Math. Res. Inst. Publ., 11, de Gruyter, Berlin, 2005.Google Scholar
  10. 10.
    D. Ginzburg, S. Rallis and D. Soudry, The descent map from automorphic representations of GL(n) to classical groups. World Scientific, Singapore, 2011. v+339 pp.Google Scholar
  11. 11.
    R. Gomez, D. Gourevitch and S. Sahi, Generalized and degenerate Whittaker models. Compositio Math. 153 (2017) 223–256.MathSciNetCrossRefGoogle Scholar
  12. 12.
    R. Howe, Automorphic forms of low rank. Noncommutative harmonic analysis and Lie groups, 211–248, Lecture Notes in Math., 880, Springer, 1981.Google Scholar
  13. 13.
    R. Howe and I. Piatetski-Shapiro, A counterexample to the “generalized Ramanujan conjecture” for (quasi-) split groups. Automorphic forms, representations and L-functions, Part 1, pp. 315–322, Proc. Sympos. Pure Math., 33, Amer. Math. Soc., Providence, R.I., 1979.Google Scholar
  14. 14.
    T. Ikeda, On the lifting of elliptic cusp forms to Siegel cusp forms of degree 2n. Ann. of Math. (2) 154 (2001), no. 3, 641–681.MathSciNetCrossRefGoogle Scholar
  15. 15.
    T. Ikeda, On the lifting of automorphic representations of \({\mathrm {PGL}}_2({\mathbb {A}})\) to \({\mathrm {Sp}}_{2n}({\mathbb {A}})\) or \(\widetilde {{\mathrm {Sp}}}_{2n+1}({\mathbb {A}})\) over a totally real field. Preprint.Google Scholar
  16. 16.
    D. Jiang, Automorphic integral transforms for classical groups I: endoscopy correspondences. Automorphic forms and related geometry: assessing the legacy of I.I. Piatetski-Shapiro, 179–242, Comtemp. Math., 614, Amer. Math. Soc., Providence, RI, 2014.Google Scholar
  17. 17.
    D. Jiang, On tensor product-L-functions and Langlands functoriality. Bull. Iranian Math. Soc., 43 (2017), no. 4, 169–489.MathSciNetGoogle Scholar
  18. 18.
    D. Jiang and B. Liu, On special unipotent orbits and Fourier coefficients for automorphic forms on symplectic groups. J. Number Theory 146 (2015), 343–389.MathSciNetCrossRefGoogle Scholar
  19. 19.
    D. Jiang and B. Liu, On Fourier coefficients of certain residual representations of symplectic groups. Pacific J. of Math. Vol. 281 (2016), No. 2, 421–466.MathSciNetCrossRefGoogle Scholar
  20. 20.
    D. Jiang and B. Liu, Fourier coefficients for automorphic forms on quasisplit classical groups. Advances in the Theory of Automorphic Forms and Their L −functions, 187–208, Contemp. Math. 664, Amer. Math. Soc., Providence, RI, 2016.Google Scholar
  21. 21.
    D. Jiang and B. Liu, Arthur parameters and Fourier coefficients for automorphic forms on symplectic groups. Ann. Inst. Fourier (Grenoble), 66 (2016), no. 2, 477–519.MathSciNetCrossRefGoogle Scholar
  22. 22.
    D. Jiang and B. Liu, Double automorphic descents and irreducible modules. Preprint. 2018.Google Scholar
  23. 23.
    D. Jiang, B. Liu, and G. Savin, Raising nilpotent orbits in wave-front sets. Representation Theory 20 (2016), 419–450.MathSciNetCrossRefGoogle Scholar
  24. 24.
    D. Jiang, B. Liu, B. Xu, and L. Zhang, The Jacquet-Langlands correspondence via twisted descent. Int. Math. Res. Notices (2016) 2016 (18): 5455–5492.MathSciNetCrossRefGoogle Scholar
  25. 25.
    D. Jiang, B. Liu and L. Zhang, Poles of certain residual Eisenstein series of classical groups. Pacific J. of Math. Vol. 264 (2013), No. 1, 83–123MathSciNetCrossRefGoogle Scholar
  26. 26.
    D. Jiang and L. Zhang, Arthur parameters and cuspidal automorphic modules of classical groups. Submitted 2015.Google Scholar
  27. 27.
    D. Jiang and L. Zhang, Automorphic Integral Transforms for Classical Groups II: Twisted Descents. Accepted by Roger Howe’s 70-th birthday volume. 2016.Google Scholar
  28. 28.
    D. Jiang, B. Liu and L. Zhang, Fourier-Jacobi periods, automorphic descents and the central value of tensor product L-functions. In preparation. 2018.Google Scholar
  29. 29.
    H. Kim and P. Sarnak, Refined estimates towards the Ramanujan and Selberg conjectures. Journal of AMS. 16 (2003), 175–181.Google Scholar
  30. 30.
    S. Kudla and S. Rallis, A regularized Siegel-Weil formula: the first term identity. Ann. of Math. (2) 140 (1994), no. 1, 1–80.MathSciNetCrossRefGoogle Scholar
  31. 31.
    R. Langlands, On the functional equations satisfied by Eisenstein series. Springer Lecture Notes in Math. 544. 1976.Google Scholar
  32. 32.
    J.-S. Li, Distinguished cusp forms are theta series. Duke Math. J. 59 (1989), no. 1, 175–189.MathSciNetCrossRefGoogle Scholar
  33. 33.
    J.-S. Li, Nonexistence of singular cusp forms. Compositio Math. 83 (1992), no. 1, 43–51.MathSciNetGoogle Scholar
  34. 34.
    B. Liu, Fourier coefficients of automorphic forms and Arthur classification. Thesis (Ph.D.) University of Minnesota. 2013. 127 pp. ISBN: 978-1303-19255-5.Google Scholar
  35. 35.
    G. Lusztig and N. Spaltenstein, Induced unipotent classes. J. London Math. Soc. (2) 19 (1979), no. 1, 41–52.MathSciNetCrossRefGoogle Scholar
  36. 36.
    C. Mœglin, Formes automorphes de carré intégrable noncuspidales. Manuscripta Math. 127 (2008), no. 4, 411–467.MathSciNetCrossRefGoogle Scholar
  37. 37.
    C. Mœglin, Image des opérateurs d’entrelacements normalisés et pôles des séries d’Eisenstein. Adv. Math. 228 (2011), no. 2, 1068–1134.MathSciNetCrossRefGoogle Scholar
  38. 38.
    C. Mœglin and J.-L. Waldspurger, Le spectre residuel de GL(n). Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 4, 605–674.MathSciNetCrossRefGoogle Scholar
  39. 39.
    C. Mœglin and J.-L. Waldspurger, Spectral decomposition and Eisenstein series. Cambridge Tracts in Mathematics, 113. Cambridge University Press, Cambridge, 1995.Google Scholar
  40. 40.
    I. Piatetski-Shapiro, On the Saito-Kurokawa lifting. Invent. Math. 71 (1983), no. 2, 309–338.MathSciNetCrossRefGoogle Scholar
  41. 41.
    I. Piatetski-Shapiro and S. Rallis, A new way to get Euler products. J. reine angew. Math. 392 (1988), 110–124.MathSciNetzbMATHGoogle Scholar
  42. 42.
    P. Sarnak, Notes on the generalized Ramanujan conjectures. Harmonic analysis, the trace formula, and Shimura varieties, 659–685, Clay Math. Proc., 4, Amer. Math. Soc., Providence, RI, 2005.Google Scholar
  43. 43.
    F. Shahidi, Eisenstein series and automorphic L-functions, volume 58 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2010. ISBN 978-0-8218- 4989-7.Google Scholar
  44. 44.
    N. Spaltenstein, Classes unipotentes et sous-groupes de Borel. Lecture Notes in Mathematics, 946. Springer-Verlag, Berlin-New York, 1982.Google Scholar
  45. 45.
    J.-L. Waldspurger, Intégrales orbitales nilpotentes et endoscopie pour les groupes classiques non ramifiés. Astérisque 269, 2001.Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of MathematicsUniversity of MinnesotaMinneapolisUSA
  2. 2.Department of MathematicsPurdue UniversityWest LafayetteUSA

Personalised recommendations