Symplectic Duality of Symmetric Spaces

Let (M, 0) ⊂ C be a complex n -dimensional Hermitian symmetric space endowed with the hyperbolic form ωB . Denote by (M ∗, ω∗ B ) the compact dual of (M,ωB) , where ω ∗ B 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 diffeomorphism ΨM : M → R = C ⊂ M∗ satisfying Ψ∗ M (ω0) = ωB and Ψ ∗ M (ω∗ B ) = ω0 . 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 C . As a byproduct of the proof of Theorem 1.1 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 (see Theorem 5.4 below).