Displacing Lagrangians in the manifolds of full flags in C^3

TL;DR

This paper investigates the displaceability of Lagrangian fibers in the manifolds of full flags in C^3, demonstrating that all but one are displaceable, and explicitly identifying the unique non-displaceable fiber.

Contribution

It applies coadjoint actions and McDuff's probes to show most fibers are displaceable, explicitly identifying the unique non-displaceable fiber in the non-monotone case.

Findings

Most Gelfand-Tsetlin fibers are displaceable.

Identifies a unique non-displaceable fiber in the non-monotone case.

Proves the displaceability of fibers using Hamiltonian isotopies.

Abstract

In symplectic geometry a question of great importance is whether a (Lagrangian) submanifold is displaceable, that is, if it can be made disjoint from itself by the means of a Hamiltonian isotopy. In these notes we analyze the coadjoint orbits of SU(n) and their Lagrangian submanifolds that are fibers of the Gelfand-Tsetlin map. We use the coadjoint action to displace a large collection of these fibers. Then we concentrate on the case n=3 and apply McDuff's method of probes to show that "most" of the generic Gelfand-Tsetlin fibers are displaceable. "Most" means "all but one" in the non-monotone case, and means "all but a 1-parameter family" in the monotone case. In the case of non-monotone manifold of full flags we present explicitly an unique non-displaceable Lagrangian fiber (S^1)^3. This fiber was already proved to be non-displaceable in \cite{NNU}. Our contribution is in displacing other fibers and thus proving the uniqueness.