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