Special Automorphic Forms on PGSp4

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


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.


Unitary Representation Parabolic Subgroup Automorphic Form Irreducible Unitary Representation Automorphic Representation 
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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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