Coadjoint representation

In mathematics, the coadjoint representation ${\displaystyle K}$ of a Lie group ${\displaystyle G}$ is the dual of the adjoint representation. If ${\displaystyle {\mathfrak {g}}}$ denotes the Lie algebra of ${\displaystyle G}$, the corresponding action of ${\displaystyle G}$ on ${\displaystyle {\mathfrak {g}}^{*}}$, the dual space to ${\displaystyle {\mathfrak {g}}}$, is called the coadjoint action. A geometrical interpretation is as the action by left-translation on the space of right-invariant 1-forms on ${\displaystyle G}$.

The importance of the coadjoint representation was emphasised by work of Alexandre Kirillov, who showed that for nilpotent Lie groups ${\displaystyle G}$ a basic role in their representation theory is played by coadjoint orbits. In the Kirillov method of orbits, representations of ${\displaystyle G}$ are constructed geometrically starting from the coadjoint orbits. In some sense those play a substitute role for the conjugacy classes of ${\displaystyle G}$, which again may be complicated, while the orbits are relatively tractable.