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Quasi-Whittaker modules for the Schrödinger algebra

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