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.

Infinite Dimensional Lie Algebras(3rd ed., Cambridge, 1990); Moody & PateraLie Algebras with Triangular Decompositions(Wiley-Interscience, 1995); CarterLie Algebras of Finite and Affine Type(Cambridge, 2005). All treat highest weight modules. $\endgroup$ – Jim Humphreys Oct 30 '17 at 13:56Highest Weight Representations of Infinite-Dimensional Lie Algebras (1987). $\endgroup$ – M.G. Oct 30 '17 at 16:06