Topology Seminar

Topology & Geometric Group Theory Seminar

Spring 2012

1:30 – 2:30, Malott 203

Tuesday, January 31

Milena Pabiniak, Cornell

Lower bounds for Gromov width of the unitary and special orthogonal coadjoint orbits

Let G be a compact connected Lie group G, 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:

min { < \alpha_j^{\vee},\lambda > | \alpha_j is a coroot and < \alpha_j^{\vee},\lambda> >0 }.

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 regular U(n) and SO(n) coadjoint orbits, and for a class of non-regular U(n) orbits. Due to the fact that the root system for SO(2n+1) is non-simply laced, using the Gelfand-Tsetlin torus action is essential: the same procedure applied to the coadjoint action of the maximal torus would give us weaker result. In the talk I will describe the Gelfand-Tsetlin system and concentrate mostly on the cases of U(n) and SO(2n+1).



← Back to the seminar home page