Characterization of cycle domains via Kobayashi hyperbolicity

Abstract

A real form $G$ of a complex semisimple Lie group $G^C$ has only finitely many orbits in any given $G^C$-flag manifold $Z=G^C/Q$. The complex geometry of these orbits is of interest, e.g., for the associated representation theory. The open orbits $D$ generally possess only the constant holomorphic functions, and the relevant associated geometric objects are certain positive-dimensional compact complex submanifolds of $D$ which, with very few well-understood exceptions, are parameterized by the Wolf cycle domains $\Omega_W(D)$ in $G^C/K^C$, where $K$ is a maximal compact subgroup of $G$. Thus, for the various domains $D$ in the various ambient spaces $Z$, it is possible to compare the cycle spaces $\Omega_W(D)$. The main result here is that, with the few exceptions mentioned above, for a fixed real form $G$ all of the cycle spaces $\Omega_W(D)$ are the same. They are equal to a universal domain $\Omega_{AG}$ which is natural from the the point of view of group actions and which, in essence, can be explicitly computed. The essential technical result is that if $\hat \Omega $ is a $G$-invariant Stein domain which contains $\Omega_{AG}$ and which is Kobayashi hyperbolic, then $\hat \Omega =\Omega_{AG}$. The equality of the cycle domains follows from the fact that every $\Omega_W(D)$ is itself Stein, is hyperbolic, and contains $\Omega_{AG}$.