THE GROMOV WIDTH OF COADJOINT ORBITS OF THE SYMPLECTIC GROUP

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{vertical bar <alpha(v), lambda >vertical bar : alpha(v) a coroot}. 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 is an element of Lie(G)* 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.