Abstract
Let G be a complex connected semi-simple Lie group, with parabolic subgroup P. Let (P, P) be its commutator subgroup. The generalized Borel-Weil theorem on flag manifolds has an analogous result on the Dolbeault cohomology H 0,q (G/(P, P)). Consequently, the dimension of H 0,q (G/(P, P)) is either 0 or ∞. In this paper, we show that the Dolbeault operator ∂̄ has closed image, and apply the Peter-Weyl theorem to show how q determines the value 0 or ∞. For the case when P is maximal, we apply our result to compute the Dolbeault cohomology of certain examples, such as the punctured determinant bundle over the Grassmannian.