EVERY SYMPLECTIC MANIFOLD IS A (LINEAR) COADJOINT ORBIT

Paul Donato (1) , Patrick Iglesias-Zemmour (2)

1 AMU - Aix Marseille Université
2 HUJ - The Hebrew University of Jerusalem

A. We prove that every symplectic manifold is a coadjoint orbit of the group of automorphisms of the integration bundle of the symplectic manifold, acting linearly on its space of momenta. And that, for any group of periods of the symplectic form. This result generalizes the Kirilov-Kostant-Souriau theorem when the symplectic manifold is homogeneous under the action of a Lie group, and the symplectic form is integral. I It is well known since Kostant, Souriau and Kirillov [Kos70] [Sou70] [Kir74], that a symplectic manifold (X, ω), homogeneous under the action of a Lie group, is isomorphic-up to a covering-to a coadjoint orbit, possibly a ne. It is less known that any symplectic manifold 1 is isomorphic to a coadjoint orbit of its group of symplectomorphisms (or Hamiltonian di feomorphisms), possibly a ne [PIZ16]. This has been established, in particular, in the rigorous framework of di feology and uses essentially the notion of Moment Map for that category [PIZ10]. But this theorem still seems to lack something. Although this is not a fundamental aw, we would like to get rid of the a ne action, de ned by a twisted cocycle of the automorphisms, and prefer to identify the symplectic manifold with an ordinary coadjoint orbit, that is an orbit of the usual linear coadjoint action 2. That could be done by integrating the a ne cocycle in some extension of the group of automorphisms rst, and then identify the a ne coadjoint orbit with an ordinary coadjoint orbit of this extension. This is quite a standard procedure in ordinary di ferential geometry, when it is possible. But we recall that we are no more in the classical framework but in di feology, and we shall see that the di culty to integrate this cocycle in an extension of the automorphisms is absorbed in this category by the ability to treat irrational toruses. The fundamental element is the integration bundle existing for any symplectic manifold, as it has been Date: November 26, 2019. 1991 Mathematics Subject Classification. 53D05, 58B99.

Mots clés: Diffeology; Symplectic Geometry; Quantization

Domaines: Géométrie différentielle [math.DG]

Paul Donato, Patrick Iglesias-Zemmour. EVERY SYMPLECTIC MANIFOLD IS A (LINEAR) COADJOINT ORBIT. 2019. ⟨hal-02401534⟩