Maass lift (Saito-Kurokawa lift) is a map which associates a degree two Siegel modular form to a Jabobi form of index 1.
Even though this map is defined by Hecke operators, one can explicitly construct it using the Fourier expansion of the given Jacobi form and indeed this construction is used to prove the modularity of the resulting form. For Siegel modular forms Koecher principle holds which causes a restriction on Fourier coefficients. On the other hand Koecher property for Jacobi forms is not automatic and the definition included a corresponding condition. Maass lift does not seem to include Koecher property at least looking at the definition formally. Then what can we get if we consider Maass lift if weak Jacobi forms (Jacobi forms without Koecher property)? Borcherds article in 1995 is probably the first consideration on this problem
(even though it was on the modular forms on orthogonal groups rather than Siegel modular forms), and one can speculate that the resulting forms are meromorphic Siegel modular forms. In this talk we discuss and solve this problem in the case of full modular group.