The bisymplectomorphism group of a bounded symmetric domain

An Hermitian bounded symmetric domain in a complex vector space, given in its circled realization, is endowed with two natural symplectic forms: the flat form and the hyperbolic form. In a similar way, the ambient vector space is also endowed with two natural symplectic forms: the Fubini-Study form and the flat form. It has been shown in "Jordan preprint archives, #212" that there exists a diffeomorphism from the domain to the ambient vector space which puts in correspondence the above pair of forms. This phenomenon is called symplectic duality for Hermitian non-compact symmetric spaces.

In this article, we first give a different and simpler proof of this fact. Then, in order to measure the non-uniqueness of this symplectic duality map, we determine the group of bisymplectomorphisms of a bounded symmetric domain, that is, the group of diffeomorphisms which preserve simultaneously the hyperbolic and the flat symplectic form. This group is a semi-direct product of the compact Lie group of linear automorphisms with an infinite-dimensional Abelian group. This result appears as a kind of Schwarz lemma.

MSC classes: Primary 53D05, 58F06; Secondary 32M15, 17C10

(This paper has appeared in Transformation Groups 13, No. 2 (2008), 283--304; DOI 10.1007/s00031-008-9015-z)

Antonio J. Di Scala < antonio.discala@polito.it >

Andrea Loi < loi@unica.it >

Guy Roos < guy.roos@normalesup.org >