The Penrose Transform for Certain Non-Compact Homogeneous Manifolds of $U(n,n)$

We construct \lq\lq{the Penrose transform}\rq\rq\ as an intertwining operator between two different geometric realization of infinite dimensional representations of $U(n,n)$, namely, from the space of the Dolbeault cohomology group on a non-compact complex homogeneous manifold to the space of holomorphic functions over the bounded domain of type $AIII$. We show that the image of the Penrose transform satisfies the system $(\Cal M_k)$ of partial differential equations of order $k+1$ which we find in explicit forms. Conversely, we also prove that any solution of the system $(\Cal M_k)$ is uniquely obtained as the image of the Penrose transform, by using the theory of prehomogeneous vector spaces