Let $\mathfrak{ g}$ be a complex finite-dimensional semisimple Lie algebra and $\mathfrak{ k}$ be any $\mathrm{ sl}(2)$-subalgebra of $\mathfrak{ g}$. In this paper we prove an earlier conjecture by Penkov and Zuckerman claiming that the first derived Zuckerman functor provides an equivalence between a truncation of a thick parabolic category $\mathcal{ O}$ for $\mathfrak{ g}$ and a truncation of the category of admissible $(\mathfrak{ g}, \mathfrak{ k})-$modules. This latter truncated category consists of admissible $(\mathfrak{ g}, \mathfrak{ k})-$modules with sufficiently large minimal $\mathfrak{ k}$-type. We construct an explicit functor inverse to the Zuckerman functor in this setting. As a corollary we obtain an estimate for the global injective dimension of the inductive completion of the truncated category of admissible $(\mathfrak{ g}, \mathfrak{ k})-$modules.
Categories: math.RT