I am looking for a reference that explains the representation theory of infinite-dimensional highest weight modules of the Kac-Moody algebra in general and for $\mathfrak{u}(n)$ (Lie algebra of unitary group $U(n)$) in particular. I appreciate any help.

I am trying to understand this paper. The construction requires the understanding of the following concepts in the representation theory of simple and affine Lie algebras:

• The construction of Verma module for a general (not necessarily integral) highest-weight state;
• The character for these modules (the Weyl-Kac character formula cannot be applied for a generic non-integral highest-weight module);
• BRST reduction of affine Lie algebras;
• Quantum Drinfeld-Sokolov reduction;
• ...

I am looking for some references that explain these concepts or some detailed examples of the construction for simplest cases. I appreciate any comment.

I am looking for a reference that explains the representation theory of infinite-dimensional highest weight modules of the Kac-Moody algebra in general and for $\mathfrak{u}(n)$ in particular. I appreciate any help.

I am looking for a reference that explains the representation theory of infinite-dimensional highest weight modules of the Kac-Moody algebra in general and for $\mathfrak{u}(n)$ (Lie algebra of unitary group $U(n)$) in particular. I appreciate any help.