Symplectic duality of symmetric spaces

Abstract

Let M \subset {\complex}^n be\ud a complex n-dimensional Hermitian symmetric space endowed with\ud 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\ud is Theorem 1 where, with the aid of the theory of Jordan triple systems, we construct an explicit {\em symplectic\ud duality}, namely a diffeomorphism \Psi_M: M\rightarrow\ud {\real}^{2n}={\complex}^n\subset M^* satisfying\ud \Psi_M^*\omega_0=\omega_{hyp} and\ud \Psi_M^*\omega_{FS}=\omega_0 for the pull-back of \Psi_M, where \omega_0\ud is the restriction to M of the flat Kaehler form of\ud the Hermitian\ud positive Jordan triple system associated to M.\ud Amongst other properties of the\ud map \Psi_M, we also show that it takes (complete) complex and\ud totally geodesic submanifolds of $M$ through the origin to complex\ud linear subspaces of {\complex}^n. As a byproduct of the proof of\ud Theorem \ref{mainteor} we get an interesting characterization\ud of the Bergman form of a Hermitian\ud symmetric space in terms of its restriction to classical complex\ud and totally geodesic submanifolds passing through the origin

Symplectic maps of complex domains into complex space forms

“…(See also [4][5][6] and [8] for further properties of McDuff's symplectomorphism.) The aim of this paper is to give an answer to Question 2 in terms of the Kähler potential of the Kähler metric of complex domains (open and connected) M ⊂ C n equipped with a Kähler form ω which admits a rotation invariant Kähler potential.…”

The diastatic exponential of a symmetric space

“…In [5] this theory has been the main tool to study the link between the symplectic geometry of an Hermitian symmetric space (M, ω hyp ) and its dual (M * , ω F S ) where ω hyp (resp. ω F S ) is the Kähler form associated to g hyp (resp.…”

The Bisymplectomorphism Group of a Bounded Symmetric Domain

“…The following theorem is the main result of [1]. We give here a different and simpler proof, using the expression of the symplectic forms ω 0 , ω − , ω + in generalized polar coordinates.…”

“…In a similar way, the ambient vector space V is also endowed with two natural symplectic forms: the Fubini-Study form ω + and the flat form ω 0 (see Section 1 for the definition of ω − , ω 0 , ω + ). It has been shown in [1] that there exists a diffeomorphism F : Ω → V such that…”

“…In Section 2 we compute the symplectic forms ω 0 and ω − using spectral decomposition in Jordan triples, which is the appropriate generalization of polar coordinates. From this, we derive in Section 3 a simple proof of property (0.1), different from the proof given in [1]. Section 4 is devoted to the study and characterization of bisymplectomorphisms.…”