Lower bounds for Gromov width of coadjoint orbits in U(n)

TL;DR

This paper establishes a new lower bound for the Gromov width of certain coadjoint orbits of the unitary group using integrable systems and symplectic embeddings, extending known results to nongeneric cases.

Contribution

It introduces a novel lower bound for Gromov width of coadjoint orbits with at most one repeated eigenvalue, generalizing previous results to nongeneric orbits.

Findings

Lower bound for Gromov width equals minimum eigenvalue difference for specific orbits.

Constructs symplectic embeddings using Gelfand-Tsetlin patterns and Hamiltonian torus actions.

Extends known Gromov width results to nongeneric coadjoint orbits.

Abstract

We use the Gelfand-Tsetlin pattern to construct an effective Hamiltonian, completely integrable action of a torus T on an open dense subset of a coadjoint orbit of the unitary group. We then identify a proper Hamiltonian T-manifold centered around a point in the dual of the Lie algebra of T. A theorem of Karshon and Tolman says that such a manifold is equivariantly symplectomorphic to a particular subset of R^2D. This fact enables us to construct symplectic embeddings of balls into certain coadjoint orbits of the unitary group, and therefore obtain a lower bound for their Gromov width. Using the identification of the dual of the Lie algebra of the unitary group with the space of (n x n) Hermitian matrices, the main theorem states that for a coadjoint orbit through \lambda=diag(\lambda_1, ..., \lambda_n) in the dual of the Lie algebra of the unitary group, where at most one eigenvalue is repeated, the lower bound for Gromov width is equal to the minimum of the differences \lambda_i-\lambda_j, over all \lambda_i>\lambda_j. For a generic orbit (i.e. with distinct \lambda_i's), with additional integrality conditions, this minimum has been proved to be exactly the Gromov width of the orbit. For nongeneric orbits this lower bound is new.