On the Structure of the Fundamental Series of Generalized Harish-chandra Modules

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 ⊂ 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 ≃ 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. Mathematics Subject Classification (2000). Primary 17B10, 17B55.