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Special Automorphic Forms on PGSp4

  • I. I. Piatetski-Shapiro
Part of the Progress in Mathematics book series (PM, volume 35)

Abstract

In a classical situation special automorphic forms were studied by Maass. Let us recall their definition. Denote by H the Siegel half plane of genus 2. Consider Siegel’s modular forms of a given weight with respect to the Siegel full modular group. It is known that they have the following Fourier decomposition:
$$f(Z)\, = \,\sum {{a_T}} \,\exp \,2\pi itr(TZ),$$
where T runs over the matrices of the form \((\begin{array}{*{20}{c}} n \\ {r/2} \end{array}\,\begin{array}{*{20}{c}}{r/2} \\ m \end{array})\) ; n, r, mZ. Put d T = 4nmr 2, e T = (n,r, m). The Maass space (following Zagier) is the space of those f (Z)such that the coefficients a T depend only on d T and e T . The forms which lie in the Maass space do not satisfy the Ramanujan conjecture. That was one of the reasons why Maass studied these forms.

Keywords

Unitary Representation Parabolic Subgroup Automorphic Form Irreducible Unitary Representation Automorphic Representation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    D. Zagier, Sur la conjecture de Saito-Kurokawa (D’apres H. Maass), Seininaire de Théorie des Nombres, Paris 1979–80. Séminaire Délange-Pisot-Poitou. Birkhiiuser.Google Scholar
  2. [2]
    H. Maass, Über eine Spezialschar von Modulformen zweiten Grades. Invent. Math. 52 (1979), 95–104.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    I. I. Piatetski-Shapiro, On the Saito-Kurokawa lifting. Preprint. February 1981.Google Scholar
  4. [4]
    M. F. Novodvorsky and I. I. Piatetski-Shapiro, Generalized Bessel models for the symplectic group of rank 2, Mat. Sb. 90 (2) (1973), 246–256.MathSciNetGoogle Scholar
  5. [5]
    I. I. Piatetski-Shapiro, L-functions for GSp( 4Preprint.Google Scholar
  6. [6]
    I. M. Gelfand and N. Ya. Vilenkin, Generalized functions, Vol. 4, Academic Press, 1964.Google Scholar
  7. [7]
    R. Howe and C. Moore, Asymptotic properties of unitary representations. J. Fun. Anal. 32 (1979), 72–96.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    R. Howe, O series and invariant theory, Proc. Symp. Pure Math., XXXIII, Part I, 275–286.Google Scholar
  9. [9]
    R. Howe, The notion of rank for representations. Preprint.Google Scholar
  10. [10]
    R. Howe and I. I. Piatetski-Shapiro, Some examples of automorphic forms on GSp 4. Preprint.Google Scholar

Copyright information

© Springer Science+Business Media New York 1983

Authors and Affiliations

  • I. I. Piatetski-Shapiro
    • 1
  1. 1.Department of MathematicsYale UniversityNew HavenUSA

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