On parametrization of symplectic quotient of Cartesian product of coadjoint orbits of complex general linear group with respect to its diagonal action

A problem of the coordinatization of the manifold constructed via the Marsden–Weinstein quotient is considered. Rational canonical coordinates on the symplectic reduction with respect to the diagonal action of the general linear group on the Cartesian product of coadjoint orbits in the case of the complex general linear group are constructed. The coordinates on the algebraically-open subset of the quotient space are presented. The method is based on the iteration process used for the construction of the projection-flag coordinates, and works if the matrices forming the orbits have a rich enough set of the invariant subspaces.