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.

It is proved that the fiber ℰ of the projection is a linear symplectic space and the map \(O\left( J \right)\tilde \to E \times O\left( {\tilde J} \right)\) is a birational symplectomorphism. Successively projecting the resulting transformations along eigensubspaces yields an isomorphism between O(J) and the linear symplectic space being the direct product of all fibers of the projections. The Darboux coordinates on O(J) are pullbacks of the canonical coordinates on this linear symplectic space.

Canonical coordinates on orbits corresponding to various Jordan structures are constructed as examples.