Birational Darboux Coordinates on (Co)Adjoint Orbits of GL(N, ℂ)

Abstract

The set of all linear transformations with a fixed Jordan structure J is a symplectic manifold isomorphic to the coadjoint orbit O(J) of the general linear group GL(N, C). Any linear transformation can be projected along its eigenspace onto a coordinate subspace of complementary dimension. The Jordan structure \(\tilde J\) of the image under the projection is determined by the Jordan structure J of the preimage; consequently, the projection \(O\left( J \right) \to O\left( {\tilde J} \right)\) is a mapping of symplectic manifolds.