Isotropic submanifolds and coadjoint orbits of the Hamiltonian group

Abstract

We describe a class of coadjoint orbits of the group of Hamiltonian diffeomorphisms of a symplectic manifold $(M,\omega)$ by implementing symplectic reduction for the dual pair associated to the Hamiltonian description of ideal fluids. The description is given in terms of nonlinear Grassmannians (manifolds of submanifolds) with additional geometric structures. Reduction at zero momentum yields the identification of coadjoint orbits with Grassmannians of isotropic volume submanifolds, slightly generalizing the results in Weinstein [1990] and Lee [2009]. At the other extreme, the case of a nondegenerate momentum recovers the identification of connected components of the nonlinear symplectic Grassmannian with coadjoint orbits, thereby recovering the result of Haller and Vizman [2004]. We also comment on the intermediate cases which correspond to new classes of coadjoint orbits. The description of these coadjoint orbits as well as their orbit symplectic form is obtained in a systematic way by exploiting the general properties of dual pairs of momentum maps. We also show that whenever the symplectic manifold $(M,\omega)$ is prequantizable, the coadjoint orbits that consist of isotropic submanifolds with total volume $a\in\mathbb{Z}$ are prequantizable. The prequantum bundle is constructed explicitly and, in the Lagrangian case, recovers the Berry bundle constructed in Weinstein [1990].