Cycle Connectivity and Automorphism Groups of Flag Domains

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A flag domain D is an open orbit of a real form G 0 in a flag manifold \(Z = G/P\) of its complexification. If D is holomorphically convex, then, since it is a product of a Hermitian symmetric space of bounded type and a compact flag manifold, Aut(D) is easily described. If D is not holomorphically convex, then in previous work it was shown that Aut(D) is a Lie group whose connected component at the identity agrees with G 0, except possibly in situations which arise in Onishchik’s list of flag manifolds where \(\mathrm{Aut}(Z)^{0} =\hat{ G}\) is larger than G. In the present work the group \(\mathrm{Aut}(D)^{0} =\hat{ G}_{0}\) is described as a real form of \(\hat{G}\). Using an observation of Kollar, new and much simpler proofs of much of our previous work in the case where D is not holomorphically convex are given.