On Whittaker modules over a class of algebras similar to U(sl 2)

Abstract

Motivated by the study of invariant rings of finite groups on the first Weyl algebras A 1 and finding interesting families of new noetherian rings, a class of algebras similar to U(sl 2) was introduced and studied by Smith. Since the introduction of these algebras, research efforts have been focused on understanding their weight modules, and many important results were already obtained. But it seems that not much has been done on the part of nonweight modules. In this paper, we generalize Kostant’s results on the Whittaker model for the universal enveloping algebras U(g) of finite dimensional semisimple Lie algebras g to Smith’s algebras. As a result, a complete classification of irreducible Whittaker modules (which are definitely infinite dimensional) for Smith’s algebras is obtained, and the submodule structure of any Whittaker module is also explicitly described.