Symplectic Duality of Symmetric Spaces

Abstract

With the aid of the theory of Jordan triple systems, we construct an explicit bi-symplectomorphism between a Hermitian symmetric space of non-compact type and $\C^n$ equipped with both the flat Kaehler-form and the Fubini-Study form. Our symplectomorphism is an explicit version of the symplectomorphism between Kaehler manifolds with non positive radial curvature and $\C^n$, whose existence was proved by Dusa McDuff. As a byproduct we get an interesting characterization of the Bergman form on a Hermitian space in terms of its restriction to classical Hermitian symmetric spaces of noncompact type.