Rational Symplectic Coordinates on the Space of Fuchs Equations m × m-Case

Abstract

The symplectic reduction method is applied to obtain an explicit description of the phase space for the system of Fuchs equations. Technically, the problem resolved is the construction of the Darboux coordinates on the quotient of the product of the several coadjoint orbits of the general linear group GL(m) over the diagonal action of GL(m). Local Darboux coordinates on the quotient space are the pull-backs of Darboux coordinates on the coadjoint orbits with respect to an explicitly described birational isomorphism between the quotient space and the Cartesian product of N − 3 coadjoint orbits of dimension m(m − 1) and an orbit of dimension (m − 1)(m − 2). In a generic case of the diagonalizable matrices it gives just an isomorphism that is birational and symplectic between open, in the Zariski topology, domain of the quotient space and the Cartesian product of the orbits in question. The global phase space is obtained by gluing up a set of 2m − 1 simple divisors to the mentioned product coadjoint orbits. The method is based on the Gauss decomposition of a matrix into the product of upper-triangular, lower-triangular and diagonal matrices. It works uniformly for the orbits formed by diagonalizable or non-diagonalizable matrices. It is elaborated for the orbits of maximal dimension that is m(m − 1).