Journal of Lie Theory

The notion of Whittaker functions on quasi-split reductive groups over local fields is usually defined for non-degenerate characters of the maximal unipotent subgroups. In this paper the case of the real symplectic group of degree two is taken up. It is remarked for irreducible generic representations that if Whittaker functions in the usual sense are of moderate growth, they are always rapidly decreasing. The target of this paper is also Whittaker functions on the symplectic group of degree two for degenerate characters, which have been out of the targets in many studies. Motivated by the theory of the Fourier-Jacobi expansion of generic cusp forms, it is proved that there is no such Whittaker functions of rapidly decay for irreducible generic representations.