Canonical domains for coadjoint orbits

This paper describes two real analytic symplectomorphisms defined on appropriate dense open subsets of any coadjoint orbit of a compact semisimple Lie algebra. The first symplectomorphism sends the open dense subset to a bounded subset of a standard cotangent bundle. The second symplectomorphism has target a bounded subset of a hyperbolic coadjoint orbit of an associated non-compact semi-simple Lie algebra. Therefore, coadjoint orbits of compact Lie algebras are symplectic compactifications of domains of cotangent bundles, and are in symplectic duality with hyperbolic orbits of non-compact semisimple Lie algebras.