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Abstract. A method for constructing birational Darboux coordinates on a coadjoint orbit of the general linear group is presented. This method is based on the Gauss decomposition of a matrix in the product of an upper-triangular and a lower-triangular matrix. The method works uniformly for the orbits formed by the diagonalizable matrices of any size and for arbitrary dimensions of the eigenspaces. §0. Introduction Our aim in this paper is to present a method for canonical parametrization of an important algebraic symplectic manifold, namely, a coadjoint orbit of the complex general linear group; see [4,5,10].The problem of description of any manifold consists of two steps. First, we should construct charts. They are sets keeping the information about the local structure of the manifold. The charts should have a global structure as simple as possible. The second step is creation of the atlas. We should glue the charts in a proper way. The law of gluing is stated by the transition functions, which identify the overlapping parts of the charts. The transition functions set the global structure of the manifold.Largely, this article is devoted to the first step. We construct one chart, a Zariski open subset of the orbit. Such a domain covers the entire orbit except for several submanifolds of dimension smaller than the dimension of the orbit. The parametrization of the charts is given analytically in Theorems 2 and 3.To describe the covering, we point out what subspaces must be in general position with the coordinate subspaces. Different charts are parametrized by renumberings of the coordinates. The transition functions can be obtained by reparametrization of the domain already parametrized in the renumerated basis. We do not present these formulas: they are bulky and useless.It should be noted that the relative arrangement of the coordinate domains of the orbit should be well understood for the following reason. There are problems where we need to glue different orbits (i.e., orbits that differ by the spectral structure of matrices they involve) to one algebraic symplectic manifold. The organization of the maps in the atlases of these manifolds is similar to the organization of the maps in one orbit. As important examples, we mention the phase spaces of the systems of equations of the isomonodromic deformations [6]- [8].We identify gl(N, C) with its dual gl * (N, C) by using the nondegenerate form A, B = tr AB. Then the coadjoint orbits are identified with the adjoint orbits. Let O J be the 2010 Mathematics Subject Classification. Primary 53D05.

Abstract. A method of constructing of Darboux coordinates on a space that is a symplectic reduction with respect to a diagonal action of GL(m, C) on a Cartesian product of N orbits of coadjoint representation of GL(m, C) is presented. The method gives an explicit symplectic birational isomorphism between the reduced space on the one hand and a Cartesian product of N − 3 coadjoint orbits of dimension m(m−1) on an orbit of dimension (m−1)(m−2) on the other hand. In a generic case of the diagonalizable matrices it gives just the isomorphism that is birational and symplectic between some open, in a Zariski topology, domain of the reduced space and the Cartesian product of the orbits in question.The method is based on a Gauss decomposition of a matrix on a product of upper-triangular, lower-triangular and diagonal matrices. It works uniformly for the orbits formed by diagonalizable or not-diagonalizable matrices. It is elaborated for the orbits of maximal dimension that is m(m − 1). IntroductionA set of Fuchs equationsmay be considered as a submanifold N n=1 A (n) = 0 of the space gl(m) × · · · × gl(m) that has a natural Poisson structure coming from gl(m).Some important problems like a problem of isomonodromic deformations may be restricted on a "symplectic leaf" of this Poisson manifold, that is a submanifold on which the Poisson structure induces the symplectic structure. It is the symplectic structure that we mean in the title.The symplectic manifold in question is closely related with the well known, "standard" symplectic manifold, the orbit of coadjoint representation of the Lie group. It is well investigated class of manifolds, and the problem of their canonical parametrization dates back to Archimedes who found a symplectomorphism between sphere, that can be considered as an orbit of SU(2), and a circumscribed cylinder.In the present article we construct an explicit birational symplectic isomorphism between the symplectic leaf of the space of Fuchs equations and a Cartesian product of the coadjoint orbits (for the version of the method for 2 × 2 matrices see [1]).