Quasi-Whittaker modules for the Schr\"odinger algebra

In this paper, we construct a new class of modules for the Schr"{o}dinger algebra $\text{mS}$, called quasi-Whittaker module. Different from \cite{[ZC]}, the quasi-Whittaker module is not induced by the Borel subalgebra of the Schr"{o}dinger algebra related with the triangular decomposition, but its Heisenberg subalgebra $\text{mH}$. We prove that, for a simple $\text{mS}$-module $V$, $V$ is a quasi-Whittaker module if and only if $V$ is a locally finite $\text{mH}$-module; Furthermore, we classify the simple quasi-Whittaker modules by the elements with the action similar to the center elements in $U(\text{mS})$ and their quasi-Whittaker vectors. Finally, we characterize arbitrary quasi-Whittaker modules.