Lower bounds for Gromov width in the SO(n) coadjoint orbits

Abstract

Let G be a compact connected Lie group G and T its maximal torus. The coadjoint orbit O_{\lambda} through \lambda in the dual of the Lie algebra of T, is canonically a symplectic manifold. Therefore we can ask the question of its Gromov width. In many known cases the width is exactly the minimum over the positive results of pairing \lambda with coroots: min{< \alpha_j^{\vee},\lambda > ; \alpha_j^{\vee} is a coroot and < \alpha_j^{\vee},\lambda > is positive}. We will show that the Gromov width for regular coadjoint orbits of the special orthogonal group is at least this minimum. The proof uses the torus action coming from the Gelfand-Tsetlin system.