Irreducible representations of untwisted affine Kac-Moody algebras

Abstract

In this paper we construct a class of new irreducible modules over untwisted affine Kac-Moody algebras $\widetilde{\mathfrak{g}}$, generalizing and including both highest weight modules and Whittaker modules. These modules allow us to obtain a complete classification of irreducible $\widetilde{\mathfrak{g}}$-modules on which the action of each root vector in $\widetilde{\mathfrak{n}}_+$ is locally finite, where $\widetilde{\mathfrak{n}}_+$ is the locally nilpotent subalgebra (or positive part) of $\widetilde{\mathfrak{g}}$. The necessary and sufficient conditions for two such irreducible $\widetilde{\mathfrak{g}}$-modules to be isomorphic are also determined. In the second part of the paper, we use the "shifting technique" to obtain a necessary and sufficient condition for the tensor product of irreducible integrable loop $\widetilde{\mathfrak{g}}$-modules and irreducible integrable highest weight $\widetilde{\mathfrak{g}}$-modules to be simple. This tensor product problem was originally studied by Chari and Pressley 28 years ago.