  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 K\"ahlerian 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 tranforms, 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.
