On Categories of Admissible $(\mathfrak{g},\mathrm{sl}(2))$-Modules

Abstract

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.

21 pages, 4 figures

References in corpus (1)

• Bounded reductive subalgebras of sl(n)