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

Abstract

This paper develops a general theory of the Fourier-Jacobi expansion of cusp forms on the real symplectic group Sp(2, R) 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-Böcherer and Berndt-Schmidt, which can be referred to as the global ingredient of our study. Based on the theory by Berndt-Böcherer we generalize the classical Eichler-Zagier correspondence in the representation theoretic context.