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Abstract

This paper is a review of results on generalized Harish-Chandra modules in
the framework of cohomological induction. The main results, obtained during the
last 10 years, concern the structure of the fundamental series of
$(\mathfrak{g},\mathfrak{k})-$modules, where $\mathfrak{g}$ is a semisimple Lie
algebra and $\mathfrak{k}$ is an arbitrary algebraic reductive in
$\mathfrak{g}$ subalgebra. These results lead to a classification of simple
$(\mathfrak{g},\mathfrak{k})-$modules of finite type with generic minimal
$\mathfrak{k}-$types, which we state. We establish a new result about the
Fernando-Kac subalgebra of a fundamental series module. In addition, we pay
special attention to the case when $\mathfrak{k}$ is an eligible $r-$subalgebra
(see the definition in section 4) in which we prove stronger versions of our
main results. If $\mathfrak{k}$ is eligible, the fundamental series of
$(\mathfrak{g},\mathfrak{k})-$modules yields a natural algebraic generalization
of Harish-Chandra's discrete series modules.