  If G_0 is a real form of a complex semisimple Lie group G and Z is compact
G-homogeneous projective algebraic manifold, then G_0 has only finitely many
orbits on Z. Complex analytic properties of open G_0-orbits D (flag domains)
are studied. Schubert incidence-geometry is used to prove the Kobayashi
hyperbolicity of certain cycle space components C_q(D). Using the hyperbolicity
of C_q(D) and analyzing the action of Aut(D) on it, an exact description of
Aut(D) is given. It is shown that, except in the easily understood case where D
is holomorphically convex with a nontrivial Remmert reduction, it is a Lie
group acting smoothly as a group of holomorphic transformations on D. With very
few exceptions it is just G_0.
