Bounded generalized Harish-Chandra modules

A (g, k)-moduleMwith finite k-multiplicities is called bounded if the k-multiplicities
of M are uniformly bounded. In this lecture, I will talk about recent results of I.
Penkov and myself concerning bounded (g, k)-modules. We prove a necessary condition
for a subalgebra k to admit an infinite-dimensional simple bounded (g, k)-
module. This condition is based on the study of primitive ideals and nilpotent orbits
in the adjoint representation of g. We then provide a construction of bounded
(g, k)-modules using localization and the theory of spherical varieties.
At the end of the lecture I will explain in detail, the classification of simple
bounded (sp(4), sl(2))-modules, sl(2) being a principal subalgebra, and will write
down their characters.