On the structure of the fundamental series of generalized Harish-Chandra modules

TL;DR

This paper advances the understanding of the structure of generalized Harish-Chandra modules, showing they have finite length and exploring conditions for reconstructibility, especially when k is isomorphic to sl(2).

Contribution

It proves that fundamental series modules are of finite length and introduces notions of reconstructibility, providing new criteria and characterizations for simple modules in this context.

Findings

Fundamental series modules have finite length.

Established sufficient conditions for strong reconstructibility when k ≅ sl(2).

Computed characters of simple strongly reconstructible modules.

Abstract

We continue the study of the fundamental series of generalized Harish-Chandra modules initiated in [PZ2]. Generalized Harish-Chandra modules are (g,k)-modules of finite type where g is a semisimple Lie algebra and k \subset g is a reductive in g subalgebra. A first result of the present paper is that a fundamental series module is a g-module of finite length. We then define the notions of strongly and weakly reconstructible simple (g,k)-modules M which reflect to what extent M can be determined via its appearance in the socle of a fundamental series module. In the second part of the paper we concentrate on the case k \simeq sl(2) and prove a sufficient condition for strong reconstructibility. This strengthens our main result from [PZ2] for the case k = sl(2). We also compute the sl(2)-characters of all simple strongly reconstructible (and some weakly reconstructible) (g,sl(2))-modules. We conclude the paper by discussing a functor between a generalization of the category O and a category of (g,sl(2))-modules, and we conjecture that this functor is an equivalence of categories.