Explicit Realizations of Certain Representations of Sp(n; R) via the Penrose Transform

We consider a family of singular infinite dimensional uni-tary representations of G = Sp(n,R) which are realized as sheaf cohomology spaces on an open G-orbit D in a general-ized flag variety for the complexification of G. By parametriz-ing an appropriate space, MD, of maximal compact subvari-eties inD, 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 represen-tation into sections of a corresponding sheaf onMD. 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. 1. Introduction. In this paper, we consider a family of singular infinite dimensional unitary representations of G = Sp(n,R) which are realized on certain sheaf cohomol