Special Automorphic Forms on PGSp ₄

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, m ∈ Z. Put d _T = 4nm − r 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.