On extensions of canonical symplectic structure from coadjoint orbit of complex general linear group

The problem of the extensions of the canonical Lee–Poisson–Kirillov–Kostant symplectic structure of the coadjoint orbit of the complex general linear group is considered. The introduced method uses the concept of the flag coordinates and does not depend on the Jordan structure of matrices forming the orbit. The principal bundle associated with the fibration of the orbit over the Grassmanian of flags is constructed.