Exhaustion functions and cohomology vanishing theorems for open orbits on complex flag manifolds (Q1902213)

Let \(G_0\) be a real semisimple Lie group, and let \(R\) be a parabolic subgroup of the complexification \(G\) of \(G_0\). Consider an open \(G_0\)-orbit \(D\) in the complex flag manifold \(X= G/R\), and let \(Y\) be a maximal compact linear subvariety of \(D\). The main results of the paper are the following: 1) A parabolic subgroup \(Q\subset R\) is constructed such that the open \(G_0\)-orbits in \(W= G/Q\) are measurable and one such orbit \(\widetilde D\) maps onto \(D\) with affine fibre. 2) It is proved that \(D\) is \((s+ 1)\)-complete in the sense of Andreotti-Grauert. More precisely, the author constructs an exhaustion function \(\varphi\) on \(D\), whose Levi form has at least \(n- s\) positive eigenvalues at each point. 3) It is shown that the space of compact linear subvarieties of \(D\) is a Stein manifold.