Quasi-Whittaker modules for the n-th Schrödinger algebra

Abstract

The n-th Schrödinger algebra schn defined in [14] is the semi-direct product of the Lie algebra sl2 with the n-th Heisenberg Lie algebra hn, which generalizes the Schrödinger algebra sl2⋉h1. Let ϕ:hn→C be a nonzero Lie algebra homomorphism. A schn-module V is called quasi-Whittaker of type ϕ if V=U(schn)v, where U(schn) is the universal enveloping algebra of schn, v is a nonzero vector such that xv=ϕ(x)v for any x∈hn. In this paper, we prove that a simple schn-module V is a quasi-Whittaker module if and only if V is a locally finite hn-module. Then we classify the simple quasi-Whittaker modules of ϕ, according to the rank of ϕ. Furthermore, we characterize arbitrary quasi-Whittaker modules through the rank of ϕ.