On Old and New Jacobi Forms

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 oldand new-forms. 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 p-adic Jacobi group. Introduction While laying the foundations for the representation theory of the Jacobi group G in [1], it has always been the policy to consider only representations of a fixed central character. For instance, both in a local or global context, all irreducible representations π of G with a fixed central character, indexed by a number m, are in bijection with the irreducible representations π̃ of the metaplectic group via the fundamental relation π = π̃ ⊗ π SW . Here π SW is the Schrödinger–Weil representation of G J . We refer to chapter 2 of [1] for the fundamentals of this theory. On the other hand, on page 41 of Eichler and Zagier [5] one can find the definition of two Hecke operators changing the level of classical Jacobi forms: Ud : Jk,m 7−→ Jk,md2 , Vd : Jk,m 7−→ Jk,md. (1) They are the analogues of the maps F (z) 7→ F (dz) for elliptic modular forms F , and are therefore the basis for the theory of Jacobi oldand newforms. For elliptic modular forms, the theory of oldand newforms is completely unvisible on the level of representations: The modular forms F (z) and F (dz) generate the same automorphic GL(2)-representation (provided F is an eigenform). However, the analogous statement for Jacobi forms is not true, since the central character of the resulting G representation is directly connected with the index of a Jacobi eigenform. This indicates that the operators Ud and Vd above should at least partly correspond to manipulations on representations. The purpose of this paper is to demonstrate how this can be accomplished. In the first section we introduce the Jacobi group as a subgroup of GSp(4). The obvious fact that the normalizer of G is more than G itself yields some automorphisms of the Jacobi group affecting the center. This simple observation leads to the definition of operators Us and Vs defined on equivalence classes of representations. Since they affect the index (i.e., the central character) of a representation, we call them index shifting operators. In section 2 we examine the effect of these operators on the