Gromov width of non-regular coadjoint orbits of U(n), SO(2n) and SO(2n+1)

TL;DR

This paper investigates the Gromov width of non-regular coadjoint orbits of U(n), SO(2n), and SO(2n+1), establishing lower bounds using torus actions from Gelfand-Tsetlin systems.

Contribution

It provides new lower bounds for the Gromov width of certain coadjoint orbits, extending known results to non-regular cases for these classical groups.

Findings

Gromov width is at least the minimum over certain coroot pairings.

Results apply to most coadjoint orbits of U(n) and SO(2n+1).

Uses Gelfand-Tsetlin torus actions in the proof.

Abstract

Let G be a compact connected Lie group G and T its maximal torus. The coadjoint orbit O_lambda through lambda in Lie(T)^* is canonically a symplectic manifold. Therefore we can ask the question about its Gromov width. In many known cases the Gromov width is exactly the minimum over the set {< alpha_j^{\vee},lambda > ; alpha_j^{\vee} a coroot and < alpha_j^{\vee},lambda > positive}. We show that the Gromov width of coadjoint orbits of the unitary group and of most of the coadjoint orbits of the special orthogonal group is at least the above minimum. The proof uses the torus action coming from the Gelfand-Tsetlin system.