Nonlinear Grassmannians: plain, decorated, augmented
Stefan Haller, Cornelia Vizman
TL;DR
This work provides a unified, functorial framework for parametric families of coadjoint orbits of classical diffeomorphism groups by using decorated and augmented nonlinear Grassmannians. It demonstrates that, under suitable good-orbit and good-isotropy conditions, these parameter spaces inherit smooth Fréchet manifold structures and carry smooth, equivariant Kirillov–Kostant–Souriau symplectic forms via moment maps, with multiple dual-pair realizations connecting to fluid and geometric mechanics. The construction encompasses isotropic and volume- (including exact-volume) preserving cases, as well as contact and associated bundle automorphism groups, and shows how decorations (densities, forms, cohomology data, orientations) and augmentations (ambient- vs submanifold-structures) yield uniform descriptions of known coadjoint orbits. The framework further shows how decorations can be treated as augmentations, ensuring compatibility of smooth structures across both viewpoints and linking to symplectic reduction in known dual-pair settings. Overall, the paper provides a systematic, broad toolkit for modeling and analyzing infinite-dimensional coadjoint orbits in a variety of geometric settings.
Abstract
Decorated and augmented nonlinear Grassmannians can be used to parametrize coadjoint orbits of classical diffeomorphism groups. We provide a general framework for decoration and augmentation functors that facilitates the construction of a smooth structure on decorated or augmented nonlinear Grassmannians. This permits to equip the corresponding coadjoint orbits with the structure of a smooth symplectic Frechet manifold. The coadjoint orbits obtained in this way are not new. Here, we provide a uniform description of their smooth structures.