KAHLER STRUCTURES AND CONVEXITY PROPERTIES OF COADJOINT ORBITS

Let g be a finite dimensional real Lie algebra and g its dual. We sho w that every coadjoint orbit in g which is contained in the relative interior of a closed convex invariant set which contains no lines carr ies a uniquely determined Kahler structure which is compatible with th e natural symplectic structure. We also characterize those orbits by d ata associated to a root decomposition with respect to a compactly emb edded Cartan algebra and characterize them among the other Kahler orbi ts meeting the dual of a compactly embedded Cartan algebra.