Generalized Harish-Chandra Modules: A New Direction

Let $\mathfrak{g}$ be a reductive Lie algebra over $\mathbf{C}$. We say that a $\frak g$-module$M$isageneralizedHarish-Chandramoduleif,forsomesubalgebra$\frak k \subset\frak g$,$M$islocally$\frak k$-finiteandhasfinite$\frak k$-multiplicities. We believe that the problem of classifying all irreducible generalized Harish-Chandra modules could be tractable. In this paper, we review the recent success with the case when $\mathfrak{k}$ is a Cartan subalgebra. We also review the recent determination of which reductive in $\mathfrak{g}$ subalgebras $\mathfrak{k}$ are essential to a classification. Finally, we present in detail the emerging picture for the case when $\mathfrak{k}$ is a principal 3-dimensional subalgebra.