In mathematics, the coadjoint representation of a Lie group is the dual of the adjoint representation. If denotes the Lie algebra of, the corresponding action of on, the dual space to, is called the coadjoint action. A geometrical interpretation is as the action by left-translation on the space of right-invariant 1-forms on. The importance of the coadjoint representation was emphasised by work of Alexandre Kirillov, who showed that for nilpotent Lie groups a basic role in their representation theory is played by coadjoint orbits. In the Kirillov method of orbits, representations of are constructed geometricallystarting from the coadjoint orbits. In some sense those play a substitute role for the conjugacy classes of, which again may be complicated, while the orbits are relatively tractable.
Formal definition
Let be a Lie group and be its Lie algebra. Let denote the adjoint representation of. Then the coadjoint representation is defined by where denotes the value of the linear functional on the vector. Let denote the representation of the Lie algebra on induced by the coadjoint representation of the Lie group. Then the infinitesimal version of the defining equation for reads: where is the adjoint representation of the Lie algebra.
A coadjoint orbit for in the dual space of may be defined either extrinsically, as the actual orbit inside, or intrinsically as the homogeneous space where is the stabilizer of with respect to the coadjoint action; this distinction is worth making since the embedding of the orbit may be complicated. The coadjoint orbits are submanifolds of and carry a natural symplectic structure. On each orbit, there is a closednon-degenerate -invariant 2-form inherited from in the following manner: The well-definedness, non-degeneracy, and -invariance of follow from the following facts: The tangent space may be identified with, where is the Lie algebra of. The kernel of the map is exactly. The bilinear form on is invariant under. is also closed. The canonical 2-form is sometimes referred to as the Kirillov-Kostant-Souriau symplectic form or KKS form on the coadjoint orbit.
