Jacobi Maass forms

In this paper, we give a new definition for the space of non-holomorphic Jacobi Maass forms (denoted by J(k,m)(nh)) of weight k is an element of Z and index m is an element of N as eigenfunctions of a degree three differential operator C-k,C-m. We show that the three main examples of Jacobi forms known in the literature: holomorphic, skew-holomorphic and real-analytic Eisenstein series, are contained in J(k,m)(nh). We construct new examples of cuspidal Jacobi Maass forms F-f of weight k is an element of 2Z and index 1 from weight k-1/2 Maass forms f with respect to Gamma(0)(4) and show that the map f bar right arrow F-f is Hecke equivariant. We also show that the above map is compatible with the well-known representation theory of the Jacobi group. In addition, we wshow that all of J(k,m)(nh) can be "essentially" obtained from scalar or vector valued half integer weight Maass forms.