ON WHITTAKER MODULES OVER A CLASS OF ALGEBRAS SIMILAR TO U(sl2)

Motivated by the study of invariant rings of finite groups on the first Weyl algebras A1 ([1]) and finding interesting families of new noetherian rings, a class of algebras similar to U(sl2) were introduced and studied by Smith in [13]. Since the introduction of these algebras, research efforts have been focused on understanding their weight modules, and many important results were already obtained in [13] and [6]. But it seems that not much has been done on the part of nonweight modules. In this note, we generalize Kostant’s results in [5] 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.