Simple Singular Whittaker Modules Over the Schrödinger Algebra

There are no simple singular Whittaker modules over most of important algebras, such as simple complex finite-dimensional Lie algebras, affine Kac–Moody Lie algebras, the Virasoro algebra, the Heisenberg–Virasoro algebra and the Schrödinger–Witt algebra. In this paper, however, we construct simple singular Whittaker modules over the Schrödinger algebra. Moreover, simple singular Whittaker modules over the Schrödinger algebra are classified. As a result, simple modules for the Schrödinger algebra which are locally finite over the positive part are completely classified. We also give characterizations of simple highest weight modules and simple singular Whittaker modules.