Fourier-Jacobi expansion of cusp forms on $Sp(2,{\Bbb R})$

TL;DR

This paper develops a comprehensive theory for the Fourier-Jacobi expansion of cusp forms on the real symplectic group of degree two, including explicit formulas and generalizations of classical correspondences.

Contribution

It provides an explicit description of Fourier-Jacobi expansions for various types of cusp forms and generalizes the Eichler-Zagier correspondence within a representation theoretic framework.

Findings

Explicit formulas for Fourier-Jacobi type spherical functions and Whittaker functions.

Generalization of the Eichler-Zagier correspondence in the context of representation theory.

Application of spectral theory for the Jacobi group to realize Fourier-Jacobi expansions.

Abstract

This paper develops a general theory of the Fourier-Jacobi expansion of cusp forms on the real symplectic group of degree two including generic cusp forms. An explicit description of such expansion is available for cusp forms generating discrete series representations, generalized principal series representations induced from the Jacobi parabolic subgroup and principal series representations, where we note that the latter two cases include non-spherical representations. As the archimedean local ingredients we need Fourier-Jacobi type spherical functions and Whittaker functions, whose explicit formulas are obtained by Hirano and by Oda, Miyazaki-Oda, Niwa and Ishii respectively. To realize these spherical functions in the Fourier-Jacobi expansion we use the spectral theory for the Jacobi group by Berndt-Boecherer and Berndt-Schmidt, which can be referred to as the global ingredient of our study. Based on the theory by Berndt-Boecherer we generalize the classical Eichler-Zagier correspondence in the representation theoretic context.