Actions symplectiques de groupes compacts. (Symplectic actions of compact groups)
The article provides a thorough and rather systematic exposition of the theory of symplectic actions of compact connected groups; complete proofs of all preliminary classical prerequisities are presented. The main result reads that the intersection of the Weyl chamber with the image of the moment map is a closed convex polyhedron and this permits to describe all symplectic actions of a torus which have at least one coisotropic orbit: they are products of a Hamiltonian coisotropic action by an antihamiltonian one. The first chapter treats symplectic manifolds, especially several improvements to the Darboux normal form \(\sum dp^i\wedge dq^i\) are presented. The second chapter deals with symplectic actions, moment mappings, holonomy groups, Hamiltonian and anti-Hamiltonian actions. The third chapter investigates the Hamiltonian actions of a torus: the image of the moment map is a convex polyhedron whose vertices are the images of the fixed points. The concluding three chapters deal with the general case of symplectic actions of compact groups and the main results mentioned above.
scientific article; zbMATH DE number 1747816
scientific article