Realizing representations on generalized flag manifolds

Abstract

Let $G$ be a complex reductive linear algebraic group and $G_0 \subseteq G$ a real form. Suppose $P$ is a parabolic subgroup of $G$ and assume that $P$ has a Levi factor $L$ such that $G_0 \cap L = L_0$ is a real form of $L$. Using the minimal globalization $V_{\min}$ of a finite length admissible representation for $L_0$, one can define a homogeneous analytic vector bundle on the $G_0$ orbit $S$ of $P$ in the generalized flag manifold $Y = G/P$. Let $A(P, V_{\min})$ denote the corresponding sheaf of polarized sections. In this article we analyze the $G_0$ representations obtained on the compactly supported sheaf cohomology groups $H^p_c(S,A(P, V_{\min}))$.