Exhaustion Functions and Cohomology Vanishing Theorems for Open Orbits on Complex Flag Manifolds

Let $G_0$ be a real semisimple Lie group, let $R$ be a parabolic subgroup of the complexification $G$ of $G_0$\,, let $D$ be an open $G_0$-orbit in the complex flag manifold $X = G/R$, and let $Y$ be a maximal compact linear subvariety of $D$. First, an explicit parabolic subgroup $Q \subset R \subset G$ is constructed so that the open $G_0$-orbits on $W = G/Q$ are measurable and one such orbit $\widetilde{D} = G_0(w) \subset W$ maps onto $D$ with affine fibre. Second, it is shown that $D$ is ($s+1$)-complete in the sense of Andreotti and Grauert, $s = \dim_\Bbb C Y$; thus cohomologies $H^q(D;\Cal F) = 0$ for $q > s$ whenever $\Cal F \to D$ is a coherent analytic sheaf. This was known \cite{7} for the case of measurable open orbits, and the proof uses that result on $\widetilde{D}$. Third, it is shown that the space $M_D$ of compact linear subvarieties of $D$ is a Stein manifold. For that, a strictly plurisubharmonic exhaustion function is constructed as in the argument \cite{9} for the case of measurable open orbits.

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Published 1 January 1995