Cycle Connectivity and Automorphism Groups of Flag Domains

Abstract

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.

Dedicated to Arkady Onishchik on the occasion of his 80th birthday.

Research for this project was supported by SFB/TR 12 and SPP 1388 of the Deutsche Forschungsgemeinschaft.