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 $\mathrm {Ad} ^{*}:G\rightarrow \mathrm {GL} ({\mathfrak {g}}^{*})$ is defined by

\langle \mathrm {Ad} _{g}^{*}\,\mu ,Y\rangle =\langle \mu ,\mathrm {Ad} _{g}^{-1}Y\rangle =\langle \mu ,\mathrm {Ad} _{g^{-1}}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 ${\mathcal {O}}_{\mu }\hookrightarrow {\mathfrak {g}}^{*}$ .