Rational symplectic coordinates on the space of Fuchs equations $m \times m$-case

TL;DR

This paper introduces a method to construct explicit Darboux coordinates on the symplectic reduction space related to Fuchsian equations, using Gauss decomposition, applicable to both diagonalizable and non-diagonalizable matrices.

Contribution

It provides a new explicit symplectic birational isomorphism between the reduced space and a product of coadjoint orbits, generalizing previous approaches for Fuchsian equations.

Findings

Constructs Darboux coordinates explicitly for the space of Fuchsian equations.

Provides a symplectic birational isomorphism applicable to diagonalizable and non-diagonalizable matrices.

Elaborates the method for orbits of maximal dimension, enhancing understanding of the symplectic structure.

Abstract

A method of constructing of Darboux coordinates on a space that is a symplectic reduction with respect to a diagonal action of $GL(m})$ on a Cartesian product of $N$ orbits of coadjoint representation of $GL(m)$ 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 $m(m-1)$.