Local newforms with global applications in the Jacobi theory (Q5929406)

The author continues his earlier investigations [Abh. Math. Sem. Univ. Hamb. 68, 273-296 (1998; Zbl 0958.11038)] on spherical representations of the Jacobi group, where he classified the good and almost good cases. In this paper he deals with the remaining bad cases.NEWLINENEWLINENEWLINETo every local spherical representation a non-negative integer is associated called the age. A local newform is a spherical representation of age 0. The precise structure of the space of spherical vectors, in particular its dimension, is determined in terms of the age. The age of the spherical principal series representations is computed, and the local newforms among these are determined. Restricting to the classical situation, it is shown that the local index shifting operators essentially coincide with well-known Hecke operators on classical modular forms which already appear in the book of \textit{M. Eichler} and \textit{D. Zagier} [The theory of Jacobi forms, Birkhäuser, Boston-Basel (1985; Zbl 0554.10018)]. This leads to some global applications of the local results.