Gromov width of a symplectic manifold $M$ is a supremum of capacities of balls that can be symplectically embedded into $M$. The definition was motivated by the Gromov's Non-Squeezing Theorem which says that maps preserving symplectic structure form a proper subset of volume preserving maps.

Let $G$ be a compact connected Lie group, $T$ its maximal torus, and $\lambda$ be a point in the chosen positive Weyl chamber.

The group $G$ acts on the dual of its Lie algebra by coadjoint action. The coadjoint orbit, $M$, through $\lambda$ 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 set of positive results of pairing $\lambda$ with coroots of $G$:

For example, this result holds if $G$ is the unitary group and $M$ is a complex Grassmannian or a complete flag manifold satisfying some additional integrality conditions.

We use the torus action coming from the Gelfand-Tsetlin system to construct symplectic embeddings of balls. In this way we prove that the above formula gives the lower bound for Gromov width of all $U(n)$ and most of $\mathrm{SO}(n)$ coadjoint orbits.

In the talk I will describe the Gelfand-Tsetlin system and concentrate mostly on the case of regular \(U(n)\) orbits.