On parametrization of phase spaces of Isomonodromic Deformations equations

The isomonodromic deformation equations, which form the main stream of the evolution of the modern theory of special functions, may be associated with some algebraic Hamiltonian system. The Hamiltonian system is defined on the space that is the symplectic quotient of the product of the several coadjoint orbits. These spaces are the subject of our investigations.

To write down the equations we should parameterize the phase space. It is non-trivial problem even in the simplest case that is the so called Painlevé VI case, where we deal with four $2\times 2$ traceless matrices and need to find two (the dimension of the Painlevé VI phase space) very special combinations of $12$(!) matrix elements. From this point of view we need to parameterize the manifold of the solutions of the system of algebraic equations on the matrix elements of $M>3$ matrices $A^k$ of $N\times N$ size.

The algebraic equations fix the Jordan forms of $A^k$ and the value of the momentum map $\sum_kA^k=0$, so the system consists of the equations of two different natures. In the generic case when the Jordan forms of $A^k$ are defined by the given eigenvalues $\lambda_i^k$, the equations are $$\det (\lambda I-A^k)=\prod_{i=1}^N(\lambda-\lambda^k_i)\ \ \forall k, \ \ \
\sum_{k=1}^MA^k=0.
$$

The parameterization functions must satisfy algebraic relations of two different natures too. They must be invariant with respect to the simultaneous similarity transformations of all $A^k$ and must have the standard values of the Lie-Poisson brackets.

In my talk I will present the transparent geometric model that makes possible to solve the problem and present the set of the rational parametrization functions. In the foundation of the construction lie an observation that for the matrix from the orbit  the projections of the kernel and image on the corresponding  complementary coordinate subspaces are conjugated each other with respect to the canonical structure of the orbit.