On Whittaker modules over a class of algebras similar to $U(s{l}_2)$

Abstract

Motivated by the study of invariant rings of finite groups on the first Weyl algebras $A_1$ (\cite{AHV}) and finding interesting families of new noetherian rings, a class of algebras similar to $U(s{l}_2)$ were introduced and studied by Smith in \cite{S}. Since the introduction of these algebras, research efforts have been focused on understanding their weight modules, and many important results were already obtained in \cite{S} and \cite{Ku}. But it seems that not much has been done on the part of nonweight modules. In this note, we generalize Kostant's results in \cite{K} on the Whittaker model for the universal enveloping algebras $U(\mathfrak{g})$ of finite dimensional semisimple Lie algebras $\mathfrak{g}$ 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.

11 pages