Dolbeault cohomology of G/(P,P)

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,H-q(G/(P, P)). Consequently, the dimension of H-0,H-q(G/(P, P)) is either 0 or infinity. In this paper, we show that the Dolbeault operator <(partial derivative)over bar> has closed image, and apply the Peter-Weyl theorem to show how q determines the value 0 or infinity. 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.