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

TL;DR

This paper proves a conjecture relating a specific functor to an equivalence of categories of admissible modules for semisimple Lie algebras, providing explicit constructions and bounds on injective dimensions.

Contribution

It establishes an equivalence between truncated categories of admissible modules via the first derived Zuckerman functor and constructs an explicit inverse functor.

Findings

Proves the conjecture by Penkov and Zuckerman.

Constructs an explicit inverse functor to Zuckerman's functor.

Provides an estimate for the global injective dimension.

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.