2006/10/29 by Tang, Xin
#20G #FOS: Mathematics #Primary 17B55 #Representation Theory (math.RT) #Secondary 17B50
paper · doi:10.48550/arxiv.math/0610897
Motivated by the study of invariant rings of finite groups on the first Weyl algebras A1 (\citeAHV) and finding interesting families of new noetherian rings, a class of algebras similar to U(sl2) were introduced and studied by Smith in \citeS. Since the introduction of these algebras, research efforts have been focused on understanding their weight modules, and many important results were already obtained in \citeS and \citeKu. But it seems that not much has been done on the part of nonweight modules. In this note, we generalize Kostant's results in \citeK on the Whittaker model for the universal enveloping algebras U(\frak g) of finite dimensional semisimple Lie algebras \frak 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.