The diastatic exponential of a symmetric space
In this paper we introduce the concept of diastatic exponential in a neighborhood
of a point of a real analytic Kaehler manifold. This concept is defined in terms of Calabi's diastasis
function and it is the analogous of the usual exponential map when the diastasis is replaced by the square
of the geodesics distance. By using the theory of Hermitian positive Jordan triple systems we prove that
for every point of an Hermitian symmetric space of noncompact type there exists a globally defined
diastatic exponential, which is uniquely determined by its restriction to polydisks (a similar result holds true
in the affine chart of an Hermitian symmetric spaces of compact type).
We also provide a geometric interpretation of the symplectic duality map in terms of diastatic exponentials
(see A. J. Di Scala and A. Loi, Symplectic duality of symmetric spaces, Adv. Math. 217 (2008), 2336-2352,
see also A. J. Di Scala, A. Loi and G. Roos, The bisymplectomorphism group of a bounded symmetric domain,
Transformation Groups Vol. 13, Number 2 (2008), 283-304).
As a corollary of our analysis we prove that the symplectic duality map pulls back the reproducing kernel of an
Hermitian symmetric space of compact type to the reproducing kernel of its dual.
A. Loi < loi@unica.it >
R. Mossa < roberto.mossa@gmail.com >