A possible symplectic framework for Radon-type transforms

Our project is to define Radon-type transforms in symplectic geometry. The chosen framework consists of symplectic symmetric spaces whose canonical connection is of Ricci-type. They can be considered as symplectic analogues of the spaces of constant holomorphic curvature in Kählerian Geometry. They are characterized amongst a class of symplectic manifolds by the existence of many totally geodesic symplectic submanifolds. We present a particular class of Radon type transforms, associating to a smooth compactly supported function on a homogeneous manifold M, a function on a homogeneous space N of totally geodesic submanifolds of M, and vice versa. We describe some spaces M and N in such Radon-type duality with M a model of symplectic symmetric space with Ricci-type canonical connection and N an orbit of totally geodesic symplectic submanifolds.SCOPUS: cp.j