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

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