Rational version of Archimedes symplectomorphysm and birational Darboux coordinates on coadjoint orbit of $GL(N,C)$

TL;DR

This paper constructs a rational symplectomorphism for coadjoint orbits of GL(N,C) with fixed Jordan form, providing explicit Darboux coordinates via projections and fiber decompositions, advancing understanding of their symplectic geometry.

Contribution

It introduces a rational symplectomorphism between coadjoint orbits and linear symplectic spaces, enabling explicit Darboux coordinates through a fibered projection approach.

Findings

The fiber of the projection is a linear symplectic space.

The orbit is birationally symplectomorphic to a product of fibers and reduced orbits.

Darboux coordinates are obtained as pull-backs from canonical coordinates.

Abstract

A set of all linear transformations with a fixed Jordan structure $J$ is a symplectic manifold isomorphic to the coadjoint orbit $\mathcal O (J)$ of $GL(N,C)$. Any linear transformation may be projected along its eigenspace to (at least one) coordinate subspace of the complement dimension. The Jordan structure $\tilde J$ of the image is defined by the Jordan structure $J$ of the pre-image, consequently the projection $\mathcal O (J)\to \mathcal O (\tilde J)$ is the mapping of the symplectic manifolds. It is proved that the fiber $\mathcal E$ of the projection is a linear symplectic space and the map $\mathcal O(J) \to \mathcal E \times \mathcal O (\tilde J)$ is a birational symplectomorphysm. The iteration of the procedure gives the isomorphism between $\mathcal O (J)$ and the linear symplectic space, which is the direct product of all the fibers of the projections. The Darboux coordinates on $\mathcal O(J)$ are pull-backs of the canonical coordinates on the linear spaces in question.