Coadjoint representation
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 geometrically starting 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
 for 
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:
 for 

where  is the adjoint representation of the Lie algebra . Coadjoint orbit

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 closed non-degenerate -invariant 2-form  inherited from  in the following manner:

. The well-definedness, non-degeneracy, and -invariance of  follow from the following facts:

(i) The tangent space  may be identified with , where  is the Lie algebra of . (ii) The kernel of the map  is exactly . (iii) 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. Properties of coadjoint orbits
The coadjoint action on a coadjoint orbit  is a Hamiltonian -action with momentum map given by the inclusion . Examples

See also

Borel–Bott–Weil theorem, for  a compact group
Kirillov character formula
Kirillov orbit theory

References
Kirillov, A.A., Lectures on the Orbit Method, Graduate Studies in Mathematics, Vol. 64, American Mathematical Society, ,

External links
 

Representation theory of Lie groups
Symplectic geometry