Symplectic duality of symmetric spaces

LOI, ANDREA;DI SCALA A. J.

2008-01-01

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Abstract

Let M \subset {\complex}^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 where, with the aid of the theory of Jordan triple systems, we construct an explicit {\em symplectic duality}, namely a diffeomorphism \Psi_M: M\rightarrow {\real}^{2n}={\complex}^n\subset M^* satisfying \Psi_M^*\omega_0=\omega_{hyp} and \Psi_M^*\omega_{FS}=\omega_0 for the pull-back of \Psi_M, where \omega_0 is the restriction to M of the flat Kaehler 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 {\complex}^n. As a byproduct of the proof of Theorem \ref{mainteor} we get an interesting characterization 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.