Dolbeault cohomology of G /( P,P )

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 . Consequently, the dimension of 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. Received: September 2, 1997; in final form February 9, 1998