Generalized Harish-Chandra modules

In this lecture I will discuss recent results by I. Penkov, G. Zuckerman and
myself in the general theory of (g, k)-modules. I will state a necessary condition
on an arbitrary subalgebra k of g to admit an irreducible (g, k)-module M with
finite k-multiplicities, with the condition that k is a maximal subalgebra acting
locally finitely on M. Moreover, I will characterize the reductive part of any such
subalgebra k. These results rely on the Beilinson-Bernstein localization theorem.
I will try to illustrate on examples how the properties of (g, k)-modules depend on
the geometry of K-orbits on the flag variety G/B.
I will also discuss a classification of (g, k)-modules with sufficiently large minimal
k-type given recently by I. Penkov and G. Zuckerman. This classification uses
the Zuckerman functor and extends some of Vogan’s methods from the case of a
symmetric pair (g, k) to an arbitrary reductive pair (g, k).