Let $\mathfrak{g}$ be a finite dimensional semisimple Lie algebra over $\mathbf{C}$. A priori one might expect the representation theory of the affine Lie algebra $\widehat{\mathfrak{g}}$ (the Lie algebra of loops into $G$, up to a central extension) to be somewhat uninteresting because it is just a Kac Moody Lie algebra (so its representations are classified in terms of roots exactly as for finite dimensional Lie algebras). However, the situation is actually somewhat subtle.
One part I've been confused about for a long time is the notion of admissible representation of $\widehat{\mathfrak{g}}$. This seems to have deep relations to geometric representation theory and the theory of vertex algberas, e.g. through the work of Arakawa, Moreau, and others.
So my question is: what is a sequence of good sources to learn about admissible representations of $\widehat{\mathfrak{g}}$? Their basic properties, maybe classification, where they appear in other subjects, etc.
