Coadjoint representation

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

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

Formal definition
Let $$G$$ be a Lie group and $$\mathfrak{g}$$ be its Lie algebra. Let $$\mathrm{Ad} : G \rightarrow \mathrm{Aut}(\mathfrak{g})$$ denote the adjoint representation of $$G$$. Then the coadjoint representation $$\mathrm{Ad}^*: G \rightarrow \mathrm{Aut}(\mathfrak{g}^*)$$ is defined by
 * $$\langle \mathrm{Ad}^*_g \, \mu, Y \rangle = \langle \mu, \mathrm{Ad}_{g} Y \rangle$$ for $$g \in G, Y \in \mathfrak{g}, \mu \in \mathfrak{g}^*,$$

where $$\langle \mu, Y \rangle$$ denotes the value of the linear functional $$\mu$$ on the vector $$Y$$.

Let $$\mathrm{ad}^*$$ denote the representation of the Lie algebra $$\mathfrak{g}$$ on $$\mathfrak{g}^*$$ induced by the coadjoint representation of the Lie group $$G$$. Then the infinitesimal version of the defining equation for $$\mathrm{Ad}^*$$ reads:
 * $$\langle \mathrm{ad}^*_X \mu, Y \rangle = \langle \mu, - \mathrm{ad}_X Y \rangle = - \langle \mu, [X, Y] \rangle$$ for $$X,Y \in \mathfrak{g}, \mu \in \mathfrak{g}^*$$

where $$\mathrm{ad}$$ is the adjoint representation of the Lie algebra $$\mathfrak{g}$$.

Coadjoint orbit
A coadjoint orbit $$\mathcal{O}_\mu$$ for $$\mu$$ in the dual space $$\mathfrak{g}^*$$ of $$\mathfrak{g}$$ may be defined either extrinsically, as the actual orbit $$\mathrm{Ad}^*_G \mu$$ inside $$\mathfrak{g}^*$$, or intrinsically as the homogeneous space $$G/G_\mu$$ where $$G_\mu$$ is the stabilizer of $$\mu$$ 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 $$\mathfrak{g}^*$$ and carry a natural symplectic structure. On each orbit $$\mathcal{O}_\mu$$, there is a closed non-degenerate $$G$$-invariant 2-form $$\omega \in \Omega^2(\mathcal{O}_\mu)$$ inherited from $$\mathfrak{g}$$ in the following manner:


 * $$\omega_\nu(\mathrm{ad}^*_X \nu, \mathrm{ad}^*_Y \nu) := \langle \nu, [X, Y] \rangle, \nu \in \mathcal{O}_\mu, X, Y \in \mathfrak{g}$$.

The well-definedness, non-degeneracy, and $$G$$-invariance of $$\omega$$ follow from the following facts:

(i) The tangent space $$\mathrm{T}_\nu \mathcal{O}_\mu = \{ -\mathrm{ad}^*_X \nu : X \in \mathfrak{g}\}$$ may be identified with $$\mathfrak{g}/\mathfrak{g}_\nu$$, where $$\mathfrak{g}_\nu$$ is the Lie algebra of $$G_\nu$$.

(ii) The kernel of the map $$X \mapsto \langle \nu, [X, \cdot] \rangle$$ is exactly $$\mathfrak{g}_\nu$$.

(iii) The bilinear form $$\langle \nu, [\cdot, \cdot] \rangle$$ on $$\mathfrak{g}$$ is invariant under $$G_\nu$$.

$$\omega$$ is also closed. The canonical 2-form $$\omega$$ 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 $$(\mathcal{O}_\mu, \omega)$$ is a Hamiltonian $G$-action with momentum map given by the inclusion $$\mathcal{O}_\mu \hookrightarrow \mathfrak{g}^*$$.