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 ⊂ G a parabolic subgroup, and (P, P) the commutator subgroup.