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 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 $K:G\rightarrow \mathrm {Aut} ({\mathfrak {g}}^{*})$ is defined as $\mathrm {Ad} ^{*}(g):=\mathrm {Ad} (g^{-1})^{*}$ . More explicitly,

$\langle K(g)F,Y\rangle =\langle F,\mathrm {Ad} (g^{-1})Y\rangle$ for $g\in G,Y\in {\mathfrak {g}},F\in {\mathfrak {g}}^{*},$ 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 for $X\in {\mathfrak {g}},K_{*}(X)=-\mathrm {ad} (X)^{*}$ where $\mathrm {ad}$ is the adjoint representation of the Lie algebra ${\mathfrak {g}}$ . One may make this observation from the infinitesimal version of the defining equation for $K$ above, which is as follows :

$\langle K_{*}(X)F,Y\rangle =\langle F,-\mathrm {ad} (X)Y\rangle$ for $X,Y\in {\mathfrak {g}},F\in {\mathfrak {g}}^{*}$ . .

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 $F$ 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 $\Omega$ , there is a closed non-degenerate $G$ -invariant 2-form $\sigma _{\Omega }$ inherited from ${\mathfrak {g}}$ in the following manner. Let $B_{F}$ be an antisymmetric bilinear form on ${\mathfrak {g}}$ defined by,

$B_{F}(X,Y):=\langle F,[X,Y]\rangle ,X,Y\in {\mathfrak {g}}$ Then one may define $\sigma _{\Omega }\in \mathrm {Hom} (\Lambda ^{2}(\Omega ),\mathbb {R} )$ by

$\sigma _{\Omega }(K_{*}(X)F,K_{*}(Y)F):=B_{F}(X,Y)$ .

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 $\mathrm {Stab} (F)$ .

(ii) The kernel of $B_{F}$ is exactly $\mathrm {stab} (F)$ .

(iii) $B_{F}$ is invariant under $\mathrm {Stab} (F)$ .

$\sigma _{\Omega }$ is also closed. The canonical 2-form $\sigma _{\Omega }$ is sometimes referred to as the Kirillov-Kostant-Souriau symplectic form or KKS form on the coadjoint orbit.

The coadjoint action on a coadjoint orbit $(\Omega ,\sigma _{\Omega })$ is a Hamiltonian $G$ -action with moment map given by $\Omega \hookrightarrow {\mathfrak {g}}^{*}$ .