Hyperbolicity of cycle spaces and automorphism groups of flag domains

If $G_0$ is a real form of a complex semisimple Lie group $G$ and $Z$ is a 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 ${\cal C}_q(D)$. Using the hyperbolicity of ${\cal C}_q(D)$ and analyzing the action of ${\rm Aut}_{\cal{O}}(D)$ on it, an exact description of ${\rm Aut}_{\cal{O}}(D)$ 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 $G_0$.