Realizing representations on generalized flag manifolds

Let G be a complex reductive linear algebraic group and a real form. Suppose P is a parabolic subgroup of G and assume that P has a Levi factor L such that is a real form of L.Using the minimal globalization V_min of a finite length admissible representation for L₀, one can define a homogeneous analytic vector bundle on the G₀ orbit S of P in the generalized flag manifold . Let denote the corresponding sheaf of polarized sections. In this article we analyze the G₀ representations obtained on the compactly supported sheaf cohomology groups.