The bisymplectomorphism group of a bounded symmetric domain
Let \(\Omega\) be a Hermitian bounded circled symmetric domain in a complex vector space \(V\). The domain \(\Omega\) is endowed with two natural symplectic forms, the flat form \(\omega_ 0\) and the hyperbolic form \(\omega_ -\). A diffeomorphism \(\varphi : \Omega\to \Omega\) such that \(\varphi^ *\omega_ 0=\omega_ 0\) and \(\varphi^ *\omega_ -=\omega_ -\) is called a bisymplectomorphim. The set of bisymplectomorphims forms an infinite dimensional group. In this paper, the authors determine the group of bisymplectomorphisms of a bounded symmetric domain, which preserve simultaneously the hyperbolic and the flat symplectic form. It is shown that if \(\Omega\) is an irreducible Hermitian bounded circled symmetric domain and if \(K\) is the isotropy group of \(0\), then the analytic or \(C^\infty\) bisymplectomorphisms of \(\Omega\) are the maps \(\varphi=f\circ g\), where \(g=d\varphi(0)\in K\) and \(f\) is associated to \(v(t) = tu(t^ 2)\), with \(u: [0,1)\to S^ 1\simeq U(1)\) analytic or \(C^\infty\) and \(u(0)=1\).
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