Symplectic duality of symmetric spaces

Let M⊂Cn be a complex n-dimensional Hermitian symmetric space endowed with the hyperbolic form ωhyp. Denote by (M∗,ωFS) the compact dual of (M,ωhyp), where ωFS is the Fubini–Study form on M∗. Our first result is Theorem 1.1 where, with the aid of the theory of Jordan triple systems, we construct an explicit symplectic duality, namely a diffeomorphism satisfying and for the pull-back of ΨM, where ω0 is the restriction to M of the flat Kähler form of the Hermitian positive Jordan triple system associated to M. Amongst other properties of the map ΨM, we also show that it takes (complete) complex and totally geodesic submanifolds of M through the origin to complex linear subspaces of Cn. As a byproduct of the proof of Theorem 1.1 we get an interesting characterization (Theorem 5.3) of the Bergman form of a Hermitian symmetric space in terms of its restriction to classical complex and totally geodesic submanifolds passing through the origin.