In mathematics, the coadjoint representation
is the dual of the adjoint representation.
denotes the Lie algebra of
, 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.
{\displaystyle \mathrm {Ad} :G\rightarrow \mathrm {Aut} ({\mathfrak {g}})}
denote the adjoint representation of
denotes the value of the linear functional
denote the representation of the Lie algebra
induced by the coadjoint representation of the Lie group
Then the infinitesimal version of the defining equation for
is the adjoint representation of the Lie algebra
in the dual space
may be defined either extrinsically, as the actual orbit
, or intrinsically as the homogeneous space
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.
in the following manner: The well-definedness, non-degeneracy, and
follow from the following facts: (i) The tangent space
(ii) The kernel of the map
(iii) The bilinear form
is sometimes referred to as the Kirillov-Kostant-Souriau symplectic form or KKS form on the coadjoint orbit.
-action with momentum map given by the inclusion