Local Newforms with Global Applications in the Jacobi Theory, Original Research Article

The study of spherical representations of the Jacobi group begun in (R. Schmidt, 1998, Abh. Math. Sem. Univ. Hamburg68, 273–296) 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.