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

A bstract . Let G0 be a real semisimple Lie group, let R be a parabolic subgroup of the complexification G of G0 , let D be an open G0-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 ⊂ R ⊂ G is constructed so that the open G0-orbits on W = G/Q are measurable and one such orbit D̃ = G0(w) ⊂ 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 C Y ; thus cohomologies Hq(D;F) = 0 for q > s whenever F → D is a coherent analytic sheaf. This was known [7] for the case of measurable open orbits, and the proof uses that result on D̃. Third, it is shown that the space MD of compact linear subvarieties of D is a Stein manifold. For that, a strictly plurisubharmonic exhaustion function is constructed as in the argument [9] for the case of measurable open orbits.