Complex Geometry and Representations of Lie Groups

Let Z = G/Q be a complex flag manifold and let Go be a real form of G. Then the representation theory of the real reductive Lie group Go is intimately connected with the geometry of Go-orbits on Z. The open orbits correspond to the discrete series representations and their analytic continuations, closed orbits correspond to the principal series, and certain other orbits give the other series of tempered representations. Here I try to indicate some of that interplay between geometry and analysis, concentrating on the complex geometric aspects of the open orbits and the related representations.