  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.
