Simple weight modules for Schrödinger algebra

Let V be a simple weight module for the Schrödinger algebra S(1) but not a simple sl2-module. Let ω∈supp(V). If V is neither a highest weight module nor a lowest weight module for S(1), we prove that supp(V)=ω+Z and all non-trivial weight spaces of V have the same dimensions. Finally, we construct an example of a simple S(1)-module with infinite-dimensional weight spaces.