A dual pair for the group of volume preserving diffeomorphisms

TL;DR

This paper constructs symplectic dual pairs related to the group of volume-preserving diffeomorphisms using cotangent bundles of embedding spaces, providing new geometric descriptions of coadjoint orbits and their foliations.

Contribution

It introduces a novel symplectic dual pair framework for volume-preserving diffeomorphisms and describes coadjoint orbits via nonlinear Grassmannians and isodrastic foliations.

Findings

Describes coadjoint orbits in terms of nonlinear Grassmannians

Constructs symplectic dual pairs involving volume-preserving diffeomorphisms

Analyzes codimension one embeddings and isodrastic foliations

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.