The Gromov width of coadjoint orbits of the symplectic group

Iva Halacheva and Milena Pabiniak

Vol. 295 (2018), No. 2, 403–420

DOI: 10.2140/pjm.2018.295.403

Abstract

We prove that the Gromov width of a coadjoint orbit of the symplectic group through a regular point $\lambda$, lying on some rational line, is at least equal to:

$$min\left\{\left|\langle {\alpha }^{\vee },\lambda \rangle \right|:{\alpha }^{\vee }\mathrm{\text{ a coroot}}\right\}.$$

Together with the results of Zoghi and Caviedes concerning the upper bounds, this establishes the actual Gromov width. This fits in the general conjecture that for any compact connected simple Lie group $G$, the Gromov width of its coadjoint orbit through $\lambda \in Lie{\left(G\right)}^{\ast }$ is given by the above formula. The proof relies on tools coming from symplectic geometry, algebraic geometry and representation theory: we use a toric degeneration of a coadjoint orbit to a toric variety whose polytope is the string polytope arising from a string parametrization of elements of a crystal basis for a certain representation of the symplectic group.

Keywords

Gromov width, coadjoint orbits, toric degenerations, Okounkov bodies, crystal bases, string polytopes

Mathematical Subject Classification 2010

Primary: 20G05, 53D99

Milestones

Received: 25 June 2016

Revised: 8 August 2017

Accepted: 23 January 2018

Published: 11 April 2018