Normal bundles of cycles in flag domains

Abstract

A real semisimple Lie group G_0 embedded in its complexification G has only finitely many orbits in any G-fag manifold Z = G/Q. The complex geometry of its open orbits D (flag domains) is studied from the point of view of compact complex submanifolds C (cycles) which arise as orbits of certain distinguished subgroups. Normal bundles E of the cycles are analyzed in some detail. It is shown that E is trivial if and only if D is holomorphically convex, in fact a product of C and a Hermitian symmetric space, and otherwise D is pseudoconcave.