Local newforms with global applications in the Jacobi theory

The study of spherical representations of the Jacobi group begun in [Sch1] is continued. Using certain index shifting operators, the notion of age of such a representation is introduced, as well as the notion of a local newform. 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 amongst 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. This leads to some global applications of the local results. Introduction The Jacobi group G is a semi-direct product of SL(2) with a three-dimensional Heisenberg group. Although it is a non-reductive algebraic group, it exhibits several features which are familiar from the theory of reductive groups, especially from the GL(2)-theory. For instance, there is a sort of classical modular forms attached to the Jacobi group, the so-called Jacobi forms, for which one can develop a Hecke theory and a theory of oldand newforms along the lines of the well-known theories for elliptic modular forms; the book [EZ] is the standard reference for classical Jacobi forms. Just as for GL(2), one can reformulate (parts of) the classical theory of Jacobi forms in terms of local and global (automorphic) representations of the underlying group, which is G . In particular, one can associate to a classical Jacobi form f an automorphic representation πf of the adelized Jacobi group G(A), a procedure which is explained in the last chapter of [BeS]. Now πf can be decomposed into local components πp, πf = ⊗