Abstract
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 $$\mathscr {O}(D)\cong \mathbb {C}$$ , 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.
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