This is another question asking for references. There is an important phenomenon of correspondence between (complexes of) representations of infinite-dimensional Lie algebras with the complementary central charges. This applies to infinite-dimensional Lie algebras of a certain kind, like the Virasoro or Kac-Moody algebras.

Let me describe the Virasoro case. Consider representations on which the central element of Virasoro acts by a constant c, and also representations on which it acts by the constant 26-c. Additional conditions have to be imposed on the representations, namely, that the positive-graded part of Virasoro acts nilpotently, or some variation of this. Then there is a way to assign (contravariantly) complexes of representations with the central charge 26-c to complexes of representations with the central charge c.

The Kac-Moody case is similar, except that there is another number (depending on the Kac-Moody algebra) in place of 26. (Actually, this holds for any Lie algebra graded by the integers with finite-dimensional components, and it can be generalized even further.)

This correspondence functor was not very well defined, classically, because it sometimes assigns acyclic complexes to nonacyclic ones and vice versa. I know how to make it well-defined, and even a (covariant) equivalence of triangulated categories. This is exactly the reason why I am asking for references to any classical expositions where the problem of constructing such a functor or equivalence were, at least, raised. I myself learned about this problem from folklore.

The only references I am presently aware of are some papers by Feigin--Fuchs and Rocha-Caridi--Wallach circa 1984 where Verma modules over the Virasoro algebra were studied. The discussion there is very brief; the very fact that one obtains acyclic complexes as duals to irreducible modules is never mentioned explicitly. Are there any later and/or more detailed references?