The Zuckerman functor as an equivalence of certain categories of g-modules

Originally, the Zuckerman derived functor was introduced in order to construct specific modules for symmetric pairs of Lie algebras. In joint work with G. Zuckerman we have used similar ideas to construct (g, k)-modules where k is a not necessarily symmetric subalgebra. It turns out that the case when k is isomorphic to sl(2) is particularly interesting and non-trivial. In this context, we construct a specific category of highest weight modules and a specific category of (g, sl(2)) modules and we conjecture that the Zuckerman functor is an equivalence between these two categories.