Pseudoconcavity of flag domains: The method of supporting cycles

A flag domain of a real from $G_0$ of a complex semismiple Lie group $G$ is an open $G_0$-orbit $D$ in a (compact) $G$-flag manifold. In the usual way one reduces to the case where $G_0$ is simple. It is known that if $D$ possesses non-constant holomorphic functions, then it is the product of a compact flag manifold and a Hermitian symmetric bounded domain. This pseudoconvex case is rare in the geography of flag domains. Here it is shown that otherwise, i.e., when $\mathcal{O}(D)\cong\mathbb{C}$, the flag domain $D$ is pseudoconcave. In a rather general setting the degree of the pseudoconcavity is estimated in terms of root invariants. This estimate is explicitly computed for domains in certain Grassmannians.