On Old and New Jacobi Forms Original Research Article

Certain “index shifting operators” for local and global representations of the Jacobi group are introduced. They turn out to be the representation theoretic analogues of the Hecke operators Ud and Vd on classical Jacobi forms, which underlie the theory of Jacobi old- and newforms. Further analogues of these operators on spaces of classical elliptic cusp forms are also investigated. In view of the correspondence between Jacobi forms and elliptic modular forms, this provides some support for a purely local conjecture about the dimension of spaces of spherical vectors in representations of the image-adic Jacobi group.