Coadjoint orbits in duals of Lie algebras with admissible ideals

We analyze the symplectic structure of the coadjoint orbits of Lie groups with Lie algebras that contain admissible ideals. Such ideals were introduced by Pukanszky to investigate the global symplectic structure of simply connected coadjoint orbits of connected, simply connected, solvable Lie groups. Using the theory of symplectic reduction of cotangent bundles, we identify classes of coadjoint orbits which are vector bundles. This implies Pukanszky's earlier result that such orbits have a symplectic form which is the sum of the canonical form and a magnetic term. This approach also allows us to provide many of the essential details of Pukanszky's result regarding the existence of global Darboux coordinates for the simply connected coadjoint orbits of connected, simply connected solvable Lie groups.