The bisymplectomorphism group of a bounded symmetric domain

In a previous paper with A. Loi we introduced the so called symplectic duality between Hermitian symmetric spaces. Such duality consists in a bysimplectomorphism between an open and dense subset of a compact Hermitian symmetric space and its non-compact dual. The question about how many dualities does exists is directly related to the group of bi-symplectomorphism of a bounded symmetric domain of the complex euclidean space. In this paper we give a precise description of such group showing that its is a product of the isotropy group times the set of smooth function of the interval [0,1) to S^1. We study the group of bi-symplectomorphism of a bounded symmetric domain.