A vanishing theorem for open orbits on complex flag manifolds

Abstract:

A real reductive Lie group $G$ acts on complex flag manifolds ${G_{\mathbf {C}}}$/(parabolic subgroup). The open orbits $D = G(x)$ are precisely the homogeneous complex manifolds $G/H$, where $H$ is the centralizer of a torus. We prove that $D$ is ($(s + 1)$)-complete in the sense of Andreotti and Grauert, with $s$ = complex dimension of a maximal compact subvariety of $D$. Thus ${H^q}(D,\mathcal {F}) = 0$ for $q > s$ and any coherent sheaf $\mathcal {F} \to D$. This vanishing theorem is needed for the realization of certain unitary representations on Dolbeault cohomology groups of homogeneous vector bundles.