Darboux Coordinates on Coadjoint Orbits of Lie Algebras

Abstract

The method of constructing spectral Darboux coordinates on finite dimensional coadjoint orbits in duals of loop algebras is applied to the one pole case, where the orbit is identified with a coadjoint orbit in the dual of a finite dimensional Lie algebra. The constructions are carried out explicitly when the Lie algebra is $\frak{sl}(2,\bold R),\ \frak{sl}(3, \bold R),$ and $\frak{so}(3, \bold R)$, and for rank two orbits in $\frak{so}(n, \bold R)$. A new feature that appears is the possibility of identifying spectral Darboux coordinates associated to ``dynamical" choices of sections of the associated eigenvector line bundles; i.e. sections that depend on the point within the given orbit.