A dual pair for the group of volume preserving diffeomorphisms

Abstract

We use cotangent bundles of spaces of smooth embeddings to construct symplectic dual pairs involving the group of volume preserving diffeomorphisms. Via symplectic reduction we obtain descriptions of coadjoint orbits of this group in terms of nonlinear Grassmannians of augmented submanifolds. For codimension one embeddings these submanifolds are further constrained to the leaves of isodrastic foliations with finite codimensions.