Maass-Jacobi forms for higher rank indices

We generalize Maass-Jacobi forms for indices in Z, hence lattices of rank 1, to Maass-Jacobi forms with index a lattice of arbitrary rank. Poincaré series can be used to analyze the space of such functions. In particular, applying this technique we can prove a Zagier type duality. The dual weights coincide with the dual weights suggested for corresponding Maass-Siegel forms. A connection to the already known skew-holomorphic Jacobi forms is revealed by introducing an appropriate xi-operator. We also briefly discuss the underlying Lie algebra and its universal enveloping algebra. Here the situation is much more involved than in the rank 1 case.