Quasi-Whittaker modules for the n-th Schrödinger algebra

The n-th Schrodinger algebra schn defined in [14] is the semidirect product of the Lie algebra sl2 with the n-th Heisenberg Lie algebra hn, which generalizes the Schrodinger algebra sl2 x h1. Let & phi; : hn & RARR; C be a nonzero Lie algebra homomorphism. A schn-module V is called quasi-Whittaker of type & phi; if V = U(schn)v, where U(schn) is the universal enveloping algebra of schn, v is a nonzero vector such that xv = & phi;(x)v for any x & ISIN; hn. In this paper, we prove that a simple schn-module Vis a quasi-Whittaker module if and only if V is a locally finite hnmodule. Then we classify the simple quasi-Whittaker modules of & phi;, according to the rank of & phi;. Furthermore, we characterize arbitrary quasi-Whittaker modules through the rank of & phi;. & COPY; 2023 Elsevier Inc. All rights reserved.