We make a first step towards a classification of simple generalized Harish-Chandra
modules which are not Harish-Chandra modules or weight modules of finite type.
For an arbitrary algebraic reductive pair of complex Lie algebras(g,t),
we construct, via cohomological induction, the fundamental series F
. (p,E) of generalized Harish-Chandra modules. We then
use F. (p,E) to characterize any simple generalized
Harish-Chandra module with generic minimal t-type. More precisely, we prove
that any such simple(g,t)-module of finite type arises as the unique
simple submodule of an appropriate fundamental series module Fs
(p,E) in the middle dimension s. Under the stronger
assumption that t contains a semisimple regular element of g,
we prove that any simple(g,t)-module with generic minimal t-type
is necessarily of finite type, and hence obtain a reconstruction theorem for a class
of simple(g,t)-module which can a priori have infinite type. We also
obtain generic general versions of some classical theorems of Harish-Chandra, such
as the Harish-Chandra admissibility theorem. The paper is concluded by examples,
in particular we compute the genericity condition on a ttype for any pair
(g,t)with t about equal to sl(2).