In mathematics, the coadjoint representation of a Lie group is the dual of the adjoint representation denotes the Lie algebra, the dual space, is called the coadjoint action. A geometrical interpretation is as the action by left-translation on the space of right-invariant 1-forms.

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 are constructed geometrically starting from the coadjoint orbits. In some sense those play a substitute role for the conjugacy classes, which again may be complicated, while the orbits are relatively tractable.

Formal definition

denote the adjoint representation. Then the coadjoint representation is defined by

.

reads:

is the adjoint representation of the Lie algebra.

Coadjoint orbit

may be defined either extrinsically, as the actual orbit, or intrinsically as the homogeneous space is the stabilizer with respect to the coadjoint action; this distinction is worth making since the embedding of the orbit may be complicated.

-invariant 2-form in the following manner:

1. .

follow from the following facts:

.

.

.

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

is a G-action with momentum map.

Examples

See also

• Borel–Bott–Weil theorem 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