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.
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