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| Dual pairs of momentum maps associated to commuting actions of two finite dimensional Lie groups have a nice property: symplectic reduction for one of the groups provides coadjoint orbits of the other group. We give a version of this result that works in infinite dimensions.
We apply it to the ideal fluid dual pair due to Marsden and Weinstein, which consists of two momentum maps defined on the manifold of embeddings $Emb(S,M)$ of a manifold $S$ endowed with a volume form into a symplectic manifold $M$. We get new coadjoint orbits of the Hamiltonian group of $M$, namely spaces of isotropic submanifolds of $M$ endowed with volume forms. |
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