The generalized Borel-Weil theorem and cohomology of G/(P,P)

Abstract

Let G be a complex semi-simple Lie group. The generalized Borel-Weil theorem computes the cohomology of a compact homogeneous G-space with coefficient in homogeneous line bundle. In this paper, we give an analogous statement via Dolbeault cohomology of the non-compact space G/(P,P) for P subset of G a parabolic subgroup, and (P,P) the commutator subgroup.