Quasi-Whittaker modules for the Schrödinger algebra

In this paper, we construct a class of new modules for the Schrödinger algebra SS, called quasi-Whittaker modules. Different from [30], the quasi-Whittaker module is not induced by the Borel subalgebra of the Schrödinger algebra related with the triangular decomposition, but by its Heisenberg subalgebra HH. We prove that, a simple SS-module V is a quasi-Whittaker module if and only if V   is a locally finite HH-module. Furthermore, we classify simple quasi-Whittaker modules by central character of U(S)U(S) and their quasi-Whittaker functions. Finally, we characterize arbitrary quasi-Whittaker modules.