EXPLICIT REALIZATIONS OF CERTAIN REPRESENTATIONS OF Sp(n,R) VIA THE DOUBLE FIBRATION TRANSFORM

We consider a family of singular infinite dimensional unitary representations of G = Sp(n,R) which are realized as sheaf cohomology spaces on an open G-orbit D in a generalized flag variety for the complexification of G. By parametrizing an appropriate space, MD, of maximal compact subvarieties in D, we identify a holomorphic double fibration between D and MD which we use to define a map P , often referred to as a double fibration or Penrose transform, from the representation into sections of a corresponding sheaf on MD. Analysis of the construction of P shows that P is injective, the image of P is the kernel of a differential operator on MD and P is an intertwining map.