A dual pair for the contact group

TL;DR

This paper introduces the EPContact dual pair, a geometric framework linking contact manifolds, weighted submanifolds, and the contact group through symplectic structures and dual pairs, with applications to geodesic equations and coadjoint orbits.

Contribution

It constructs a novel infinite-dimensional symplectic structure called the EPContact dual pair, generalizing symplectization and connecting contact geometry with group actions and coadjoint orbits.

Findings

Defines the EPContact dual pair with Hamiltonian group actions

Provides a symplectic reduction linking submanifolds to coadjoint orbits

Describes singular solutions for geodesic equations on contact diffeomorphisms

Abstract

Generalizing the canonical symplectization of contact manifolds, we construct an infinite dimensional non-linear Stiefel manifold of weighted embeddings into a contact manifold. This space carries a symplectic structure such that the contact group and the group of reparametrizations act in a Hamiltonian fashion with equivariant moment maps, respectively, giving rise to a dual pair, called the EPContact dual pair. Via symplectic reduction, this dual pair provides a conceptual identification of non-linear Grassmannians of weighted submanifolds with certain coadjoint orbits of the contact group. Moreover, the EPContact dual pair gives rise to singular solutions for the geodesic equation on the group of contact diffeomorphisms. For the projectivized cotangent bundle, the EPContact dual pair is closely related to the EPDiff dual pair due to Holm and Marsden, and leads to a geometric description of some coadjoint orbits of the full diffeomorphism group.