Coadjoint representation

In mathematics, the coadjoint representation $K$ of a Lie group $G$ is the dual of the adjoint representation. If $\mathfrak$ denotes the Lie algebra of $G$, the corresponding action of $G$ on $\mathfrak^*$, the dual space to $\mathfrak$, 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 $\mathfrak$ be its Lie algebra. Let $\mathrm : G \rightarrow \mathrm(\mathfrak)$ denote the adjoint representation of $G$. Then the coadjoint representation $\mathrm^*: G \rightarrow \mathrm(\mathfrak^*)$ is defined by
:$\langle \mathrm^*_g \, \mu, Y \rangle = \langle \mu, \mathrm_ Y \rangle$ for $g \in G, Y \in \mathfrak, \mu \in \mathfrak^*,$
where $\langle \mu, Y \rangle$ denotes the value of the linear functional $\mu$ on the vector $Y$.

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

where $\mathrm$ is the adjoint representation of the Lie algebra $\mathfrak$.

Coadjoint orbit

A coadjoint orbit $\mathcal_\mu$ for $\mu$ in the dual space $\mathfrak^*$ of $\mathfrak$ may be defined either extrinsically, as the actual orbit $\mathrm^*_G \mu$ inside $\mathfrak^*$, 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 $\mathfrak^*$ and carry a natural symplectic structure. On each orbit $\mathcal_\mu$, there is a closed non-degenerate $G$-invariant 2-form $\omega \in \Omega^2(\mathcal_\mu)$ inherited from $\mathfrak$ in the following manner:

:$\omega_\nu(\mathrm^*_X \nu, \mathrm^*_Y \nu) := \langle \nu,$, Y\rangle , \nu \in \mathcal_\mu, X, Y \in \mathfrak.

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

(i) The tangent space $\mathrm_\nu \mathcal_\mu = \$ may be identified with $\mathfrak/\mathfrak_\nu$, where $\mathfrak_\nu$ is the Lie algebra of $G_\nu$.

(ii) The kernel of the map X \mapsto \langle \nu, , \cdot\rangle is exactly $\mathfrak_\nu$.

(iii) The bilinear form \langle \nu, cdot, \cdot\rangle on $\mathfrak$ 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 $(\mathcal_\mu, \omega)$ is a Hamiltonian $G$-action with momentum map given by the inclusion $\mathcal_\mu \hookrightarrow \mathfrak^*$.

Examples

See also

* Borel–Bott–Weil theorem, for $G$ 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

* {{planetmath reference, urlname=CoadjointOrbit, title=Coadjoint orbit

Representation theory of Lie groups
Symplectic geometry