ON CATEGORIES OF ADMISSIBLE ( g $$ \mathfrak{g} $$ , sl(2))-MODULES

Abstract

Let \( \mathfrak{g} \) be a complex finite-dimensional semisimple Lie algebra and \( \mathfrak{k} \) be any 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.