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 $\tilde{\mathfrak{g}}$, generalizing and including both highest weight modules and Whittaker modules. These modules allow us to obtain a complete classification of irreducible $\tilde{\mathfrak{g}}$-modules on which the action of each root vector in ${\tilde{\mathfrak{n}}}_{+}$ is locally finite, where ${\tilde{\mathfrak{n}}}_{+}$ is the locally nilpotent subalgebra (or positive part) of $\tilde{\mathfrak{g}}$. The necessary and sufficient conditions for two such irreducible $\tilde{\mathfrak{g}}$-modules to be isomorphic are also determined. Based on the first part of the paper, we develop the so-called “shifting technique”, which enables us to obtain a necessary and sufficient condition for the tensor product of irreducible integrable loop $\tilde{\mathfrak{g}}$-modules and irreducible integrable highest weight $\tilde{\mathfrak{g}}$-modules to be simple. This tensor product problem was first studied and partly solved by Chari and Pressley in 1987, and remained open since then. Our result also recovers the result for ${\widehat{\mathfrak{sl}}}_2$ by Adamović (1997).