Algebraic Methods in the Theory of Generalized Harish-Chandra Modules

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 Sect. 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.

Both authors have been partially supported by the DFG through Priority Program 1388 “Representation theory.”