The Diastatic Exponential of a Symmetric Space

= gp (v, v) , ∀v ∈ W, where Dp is Calabi’s diastasis function at p (the usual exponential expp obviously satisfied these equations when Dp is replaced by the square of the geodesics distance d2p from p). In this paper we prove that for every point p of an Hermitian symmetric space of noncompact type M there exists a globally defined diastatic exponential centered in p which is a diffeomorphism and it is uniquely determined by its restriction to polydisks. An analogous result holds true in an open dense neighbourhood of every point of M∗, the compact dual of M . We also provide a geometric interpretation of the symplectic duality map (recently introduced in [5]) in terms of diastatic exponentials. As a byproduct of our analysis we show that the symplectic duality map pulls back the reproducing kernel of M∗ to the reproducing kernel of M .