On canonical parametrization of phase spaces of Isomonodromic Deformation Equations

Abstract

The space of the Fuchsian systems is the algebraic Poisson manifold, and the equations of the Isomonodromic Deformations are the Hamiltonian equations. The internal symmetry of the problem makes it possible to reduce the dimension of the problem using the symplectic-quotient theory. The phase-space is constructed from the orbits of (co)adjoint representation of the general linear group. The presented parametrisation of the quotientspace is based on the construction of the flag coordinates on the orbits. The simplest non-trivial case that is Painlevé VI case is considered as an example.