Simple Schrödinger modules which are locally finite over the positive part

It is proved that a simple module which is locally finite over the positive part of the Cartan decomposition for a finite-dimensional Lie algebra with triangular decomposition and solvable positive part is a highest weight module, a Whittaker module or a singular Whittaker module. The classification of simple highest weight modules for the Schrödinger algebra follows from the classification of all simple lowest weight module for the Schrödinger algebra given by Dobrev, Doebner and Mrugalla. The simple Whittaker modules for the Schrödinger algebra are classified in this paper. The singular Whittaker vectors for the universal singular Whittaker module are given.