Coadjoint representation

In mathematics, the coadjoint representation $K$ of a Lie group $G$ is the dual of the adjoint representation. If $𝔤$ denotes the Lie algebra of $G$, the corresponding action of $G$ on ${𝔤}^{*}$, the dual space to $𝔤$, 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 $𝔤$ be its Lie algebra. Let $\mathrm{A}\mathrm{d}:G\to \mathrm{A}\mathrm{u}\mathrm{t}(𝔤)$ denote the adjoint representation of $G$. Then the coadjoint representation ${\mathrm{A}\mathrm{d}}^{*}:G\to \mathrm{G}\mathrm{L}({𝔤}^{*})$ is defined by

$⟨{\mathrm{A}\mathrm{d}}_g^{*}\mu ,Y⟩=⟨\mu ,{\mathrm{A}\mathrm{d}}_g^{-1}Y⟩=⟨\mu ,{\mathrm{A}\mathrm{d}}_{g^{-1}}Y⟩$ for $g\in G,Y\in 𝔤,\mu \in {𝔤}^{*},$

where $⟨\mu ,Y⟩$ denotes the value of the linear functional $\mu$ on the vector $Y$.

Let ${\mathrm{a}\mathrm{d}}^{*}$ denote the representation of the Lie algebra $𝔤$ on ${𝔤}^{*}$ induced by the coadjoint representation of the Lie group $G$. Then the infinitesimal version of the defining equation for ${\mathrm{A}\mathrm{d}}^{*}$ reads:

$⟨{\mathrm{a}\mathrm{d}}_X^{*}\mu ,Y⟩=⟨\mu ,-{\mathrm{a}\mathrm{d}}_{X}Y⟩=-⟨\mu ,[X,Y]⟩$ for $X,Y\in 𝔤,\mu \in {𝔤}^{*}$

where $\mathrm{a}\mathrm{d}$ is the adjoint representation of the Lie algebra $𝔤$.

Coadjoint orbit

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

${\omega }_{\nu }({\mathrm{a}\mathrm{d}}_X^{*}\nu ,{\mathrm{a}\mathrm{d}}_Y^{*}\nu ):=⟨\nu ,[X,Y]⟩,\nu \in {𝒪}_{\mu },X,Y\in 𝔤$.

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

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

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

(iii) The bilinear form $⟨\nu ,[\cdot ,\cdot ]⟩$ on $𝔤$ 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 $({𝒪}_{\mu },\omega )$ is a Hamiltonian $G$-action with momentum map given by the inclusion ${𝒪}_{\mu }\hookrightarrow {𝔤}^{*}$.

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, Template:ISBN, Template:ISBN

External links

• Template:Planetmath reference