Skip to main content
SpringerLink
Log in

A spectral identity between symmetric spaces

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

Let π be a cuspidal automorphic representation ofGL 2n . We prove an identity between two spectral distributions onSp 2n andGL 2n respectively. The first is the spherical distribution with respect toSp n×Sp nof the residual Eisenstein series induced from π. The second is the weighted spherical distribution of π with respect toGL n×GL nand a certain degenerate Eisenstein series. A similar identity relates the pair (U 2n ,Sp n) and (GL n/E,GL n/F) whereE/F is the quadratic extension defining the quasi-split unitary groupU 2n . We also have a Whittaker version of these trace identities.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. Ash, D. Ginzburg and S. Rallis,Vanishing periods of cusp forms over modular symbols, Mathematische Annalen296 (1993), 709–723.

    Article  MATH  MathSciNet  Google Scholar 

  2. J. Arthur,A trace formula for reductive groups. II. Applications of a truncation operator, Compositio Mathematica40 (1980), 87–121.

    MATH  MathSciNet  Google Scholar 

  3. D. Bump and S. Friedberg,The exterior square automorphic L-functions on GL(n), inFestschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part II (Ramat Aviv, 1989), Israel Mathematical Conference Proceedings, Vol. 3, Weizmann Science Press of Israel, Jerusalem, 1990, pp. 47–65.

    Google Scholar 

  4. J. Dixmier and P. Malliavin,Factorisations de fonctions et de vecteurs indéfiniment différentiables, Bulletin des Sciences Mathématiques. 2e Série102 (1978), 307–330.

    MathSciNet  Google Scholar 

  5. Y. Z. Flicker,Twisted tensors and Euler products, Bulletin de la Société Mathématique de France116 (1988), 295–313.

    MATH  MathSciNet  Google Scholar 

  6. D. Ginzburg and D. Jiang,Periods and liftings: From G 2 to C 3, Israel Journal of Mathematics123 (2001), 29–59.

    Article  MATH  MathSciNet  Google Scholar 

  7. D. Ginzburg and E. Lapid,On a conjecture of Jacquet-Lai-Rallis; some exceptional cases, preprint.

  8. D. Ginzburg, S. Rallis and D. Soudry,On explicit lifts of cusp forms from GL m to classical groups, Annals of Mathematics (2)150 (1999), 807–866.

    Article  MATH  MathSciNet  Google Scholar 

  9. Harish-Chandra,Automorphic Forms on Semisimple Lie Groups, Notes by J. G. M. Mars, Lecture Notes in Mathematics62, Springer-Verlag, Berlin, 1968.

    MATH  Google Scholar 

  10. H. Jacquet,A guide to the relative trace formula, preprint.

  11. H. Jacquet,Automorphic spectrum of symmetric spaces, inRepresentation Theory and Automorphic Forms (Edinburgh, 1996), American Mathematical Society, Providence, RI, 1997, pp. 443–455.

    Google Scholar 

  12. D. Jiang,G 2-periods and residual representations, Journal für die reine und angewandte Mathematik497 (1998), 17–46.

    Article  MATH  Google Scholar 

  13. H. Jacquet, K. F. Lai and S. Rallis,A trace formula for symmetric spaces, Duke Mathematical Journal70 (1993), 305–372.

    Article  MATH  MathSciNet  Google Scholar 

  14. H. Jacquet, E. Lapid and S. Rallis,A spectral identity for skew symmetric matrices, inContributions to Automorphic Forms, Geometry and Number Theory, Shalikafest 2002, A Supplemental Volume to the American Journal of Mathematics, 2003.

  15. D. Jiang, Z. Mao and S. Rallis,A relative Kuznietsov trace formula on G 2, Manuscripta Mathematica99 (1999), 411–423.

    Article  MATH  MathSciNet  Google Scholar 

  16. H. Jacquet and S. Rallis,Kloosterman integrals for skew symmetric matrices, Pacific Journal of Mathematics154 (1992), 265–283.

    MATH  MathSciNet  Google Scholar 

  17. H. Jacquet and S. Rallis,Symplectic periods, Journal für die reine und angewandte Mathematik423 (1992), 175–197.

    MathSciNet  Google Scholar 

  18. H. Jacquet and J. A. Shalika,On Euler products and the classification of automorphic forms. II, American Journal of Mathematics103 (1981), 777–815.

    Article  MATH  MathSciNet  Google Scholar 

  19. H. Jacquet and J. Shalika,Exterior square L-functions, inAutomorphic Forms, Shimura Varieties, and L-functions, Vol. II (Ann Arbor, MI, 1988), Academic Press, Boston, MA, 1990, pp. 143–226.

    Google Scholar 

  20. H. H. Kim and F. Shahidi,On the holomorphy of certain L-functions, inContributions to Automorphic Forms, Geometry and Number Theory, Shalikafest 2002, A Supplemental Volume to the American Journal of Mathematics, 2003.

  21. R. P. Langlands,Euler Products, A James K. Whittemore Lecture in Mathematics given at Yale University, 1967, Yale Mathematical Monographs, 1, Yale University Press, New Haven, Conn., 1971.

    Google Scholar 

  22. E. Lapid,On the spectral expansion of Jacquet’s relative trace formula, preprint.

  23. E. Lapid and J. Rogawski,Periods of Eisenstein series: The Galois case, Duke Mathematical Journal120 (2003), 153–226.

    Article  MATH  MathSciNet  Google Scholar 

  24. C. Mœglin and J.-L. Waldspurger,Spectral Decomposition and Eisenstein Series, Une paraphrase de l’Écriture [A paraphrase of Scripture], Cambridge University Press, Cambridge, 1995.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, 91904, Jerusalem, Israel

    Erez Lapid

  2. Department of Mathematics, The Ohio State University, 43210, Columbus, OH, USA

    Stephen Rallis

Authors
  1. Erez Lapid
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stephen Rallis
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

First-named author partially supported by NSF grant DMS 0070611.

Second-named author partially supported by NSF grant DMS 9970342.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lapid, E., Rallis, S. A spectral identity between symmetric spaces. Isr. J. Math. 140, 221–244 (2004). https://doi.org/10.1007/BF02786633

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02786633

Keywords

Search

Navigation