Semisimple Symplectic Symmetric Spaces

Abstract

A symplectic symmetric space is a connected affine symmetric manifold M endowed with a symplectic structure ω which is invariant under the geodesic symmetries. When the transvection group G0 of such a symmetric space M is semisimple, its action on (M,ω) is strongly Hamiltonian; a classical theorem due to Kostant implies that the moment map associated to this action realises a G0-equivariant symplectic covering of a coadjoint orbit O in the dual of the Lie algebra $$\mathcal{G}^0 $$ of G0. We show that this orbit itself admits a structure of symplectic symmetric space whose transvection algebra is $$\mathcal{G}^0 $$ . The main result of this paper is the classification of symmetric orbits for any semisimple Lie group. The classification is given in terms of root systems of transvection algebras and therefore provides, in a symplectic framework, a theorem analogous to the Borel–de Siebenthal theorem for Riemannian symmetric spaces. When its dimension is greater than 2, such a symmetric orbit is not regular and, in general, neither Hermitian nor pseudo-Hermitian.