Fourier-Jacobi expansion of cusp forms on $Sp$(2;$mathbb{R}$）(Analytic, geometric and $p$-adic aspects of automorphic forms and $L$-functions)

Abstract

This note announces the recent result by the author about a general theory of the Fourier-Jacobi expansion of cusp forms on Sp(2; R). It is also viewed as the write-up of author's talk at the RIMS workshop in January 2020. The theory covers the case of generic cusp forms. Explicit descriptions of such expansion are available for cusp forms generating large discrete series representations, generalized principal series representations induced from a Jacobi parabolic subgroup and principal series representations (induced from the minimal parabolic subgroup), which are known to be generic. As the archimedean local ingredients we need the notion of Fourier-Jacobi type spherical functions and Whittaker functions, whose explicit formulas are obtained by Hirano and by Oda, Miyazaki-Oda, Niwa and Ishii et al. 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. This note includes the correction to author's presentation at the workshop. The Fourier-Jacobi expansion has some contribution by Eisenstein-Poincaré series with the test functions given by the Whittaker functions, for which the author had completely no idea when he gave the talk.