Symplectic duality of symmetric spaces

Let M subset of C-n be a complex n-dimensional Hermitian symmetric space endowed with the hyperbolic form omega(hyp). Denote by (M*, omega(FS)) the compact dual of (M, omega(hyp)), where omega(FS) 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 Psi(M):M -> R-2n = C-n subset of M* satisfying Psi(*)(M)omega(0) = omega(hyp) and Psi(*)(M)omega(FS) = omega(0) for the pull-back of PM, where omega(0) is the restriction to M of the flat Kahler form of the Hermitian positive Jordan triple system associated to M. Amongst other properties of the map Psi(M), we also show that it takes (complete) complex and totally geodesic submanifolds of M through the origin to complex linear subspaces of C-n. 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. (c) 2007 Elsevier Inc. All rights reserved.