Algebraic methods in the theory of generalized Harish-Chandra modules

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 (g, k)−modules, where g is a semisimple Lie algebra and k is an arbitrary algebraic reductive in g subalgebra. These results lead to a classification of simple (g, k)−modules of finite type with generic minimal 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 k is an eligible r−subalgebra (see the definition in section 4) in which we prove stronger versions of our main results. If k is eligible, the fundamental series of (g, k)−modules yields a natural algebraic generalization of Harish-Chandra’s discrete series modules. Mathematics Subject Classification (2010). Primary 17B10, 17B55.