Coadjoint representation
In mathematics, the coadjoint representation formula_1 of a Lie group formula_2 is the dual of the adjoint representation. If formula_3 denotes the Lie algebra of formula_2, the corresponding action of formula_2 on formula_6, the dual space to formula_3, is called the coadjoint action. A geometrical interpretation is as the action by left-translation on the space of right-invariant 1-forms on formula_2. The importance of the coadjoint representation was emphasised by work of Alexandre Kirillov, who showed that for nilpotent Lie groups formula_2 a basic role in their representation theory is played by coadjoint orbits. In the  Kirillov method of orbits, representations of formula_2 are constructed geometrically starting from the coadjoint orbits. In some sense those play a substitute role for the conjugacy classes of formula_2, which again may be complicated, while the orbits are relatively tractable. Formal definition. Let formula_2 be a Lie group and formula_3 be its Lie algebra. Let formula_14 denote the adjoint representation of formula_2. Then the coadjoint representation formula_16 is defined by where formula_19 denotes the value of the linear functional formula_20 on the vector formula_21. Let formula_22 denote the representation of the Lie algebra formula_3 on formula_6 induced by the coadjoint  representation of the Lie group formula_2. Then the infinitesimal version of the defining equation for formula_26 reads: where formula_29 is the adjoint representation of the Lie algebra formula_3. Coadjoint orbit. A coadjoint orbit formula_31 for formula_20 in the dual space formula_6 of formula_3 may be defined either extrinsically, as the actual orbit formula_35 inside formula_6, or intrinsically as the homogeneous space formula_37 where formula_38 is the stabilizer of formula_20 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 formula_6 and carry a natural  symplectic structure. On each orbit formula_31, there is a closed non-degenerate formula_2-invariant 2-form formula_43 inherited from formula_3 in the following manner: The well-definedness, non-degeneracy, and formula_2-invariance of formula_47 follow from the following facts: (i) The tangent space formula_48 may be identified with formula_49, where formula_50 is the Lie algebra of formula_51. (ii) The kernel of the map formula_52 is exactly formula_50. (iii) The bilinear form formula_54 on formula_3 is invariant under formula_51. formula_47 is also closed. The canonical 2-form formula_47 is sometimes referred to as the "Kirillov-Kostant-Souriau symplectic form" or "KKS  form" on the coadjoint orbit. Properties of coadjoint orbits. The coadjoint action on a coadjoint orbit formula_59 is a Hamiltonian formula_2-action with momentum map given by the inclusion formula_61.