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 dual 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 orbit. 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 $K:G\rightarrow {\mathfrak {g}}^{*}$ is defined as $\mathrm {Ad} ^{*}(g^{-1}):=\mathrm {Ad} (g^{-1})^{*}$ . More explicitly, $\langle K(g)F,X\rangle =\langle F,\mathrm {Ad} (g^{-1})X\rangle$ for $g\in G,F\in {\mathfrak {g}}^{*},X\in {\mathfrak {g}}$ where $\langle F,Y\rangle$ denotes the value of a linear functional $F$ on a vector $Y$ .

Let $K_{*}$ denote the representation of the Lie algebra ${\mathfrak {g}}$ on ${\mathfrak {g}}^{*}$ induced by the coadjoint representation of the Lie group $G$ . Then $K_{*}=-\mathrm {ad} ^{*}$ where $\mathrm {ad}$ is the adjoint representation of the Lie algebra ${\mathfrak {g}}$ . One may see that from the infinitesimal version of the defining equation for $K$ above, which is $\langle K_{*}(X)F,Y\rangle =\langle F,-\mathrm {ad} (X)Y\rangle$ for $X,Y\in {\mathfrak {g}},F\in {\mathfrak {g}}^{*}$ . .

Coadjoint orbit

A coadjoint orbit $\Omega :={\mathcal {O}}(F)$ for $F$ in the dual space ${\mathfrak {g}}^{*}$ of ${\mathfrak {g}}$ may be defined either extrinsically, as the actual orbit $K(G)(F)$ inside ${\mathfrak {g}}^{*}$ , or intrinsically as the homogeneous space $G/\mathrm {Stab} (F)$ where $\mathrm {Stab} (F)$ is the stabilizer of $F$ ; this distinction is worth making since the embedding of the orbit may be complicated.

The coadjoint orbits are all symplectic manifolds. On each orbit $\Omega$ , there is a natural closed non-degenerate $G$ -invariant 2-form $\sigma _{\Omega }$ inherited from ${\mathfrak {g}}$ in the following manner. Let $B_{F}(X,Y):=\langle F,[X,Y]\rangle ,X,Y\in {\mathfrak {g}}$ be an antisymmetric bilinear form on ${\mathfrak {g}}$ . Then $\sigma _{\Omega }$ may be defined by $\sigma _{\Omega }(F)(K_{*}(X)(F),K_{*}(Y)(F)):=B_{F}(X,Y)$ . The well-definedness, non-degeneracy, and $G$ -invariance of $\sigma _{\Omega }$ follow from the following facts :

(i) The tangent space $T_{F}(\Omega )$ may be identified with ${\mathfrak {g}}/\mathrm {stab} (F)$ , where $\mathrm {stab} (F)$ is the Lie algebra of $\mathrm {Stab} (F)$ .