The bisymplectomorphism group of a bounded symmetric domain

TL;DR

This paper studies the group of diffeomorphisms that preserve two natural symplectic forms on a bounded symmetric domain, revealing its structure as a product of a compact Lie group and an infinite-dimensional Abelian group.

Contribution

It provides a simpler proof of symplectic duality and characterizes the bisymplectomorphism group of bounded symmetric domains, detailing its structure.

Findings

The bisymplectomorphism group is the product of a compact Lie group and an infinite-dimensional Abelian group.

A new, simpler proof of symplectic duality for Hermitian bounded symmetric domains.

The structure of the bisymplectomorphism group acts as a Schwarz lemma.

Abstract

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 arXiv:math.DG/0603141 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 the 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.