In mathematics, the coadjoint representation ${\displaystyle K}$ of a Lie group ${\displaystyle G}$ is the dual of the adjoint representation. If ${\displaystyle {\mathfrak {g}}}$ denotes the Lie algebra of ${\displaystyle G}$, the corresponding action of ${\displaystyle G}$ on ${\displaystyle {\mathfrak {g}}^{*}}$, the dual space to ${\displaystyle {\mathfrak {g}}}$, is called the coadjoint action. A geometrical interpretation is as the action by left-translation on the space of right-invariant 1-forms on ${\displaystyle G}$.

The importance of the coadjoint representation was emphasised by work of Alexandre Kirillov, who showed that for nilpotent Lie groups ${\displaystyle G}$ a basic role in their representation theory is played by coadjoint orbit. In the Kirillov method of orbits, representations of ${\displaystyle G}$ are constructed geometrically starting from the coadjoint orbits. In some sense those play a substitute role for the conjugacy classes of ${\displaystyle G}$, which again may be complicated, while the orbits are relatively tractable.

Formal definition

Let ${\displaystyle G}$ be a Lie group and ${\displaystyle {\mathfrak {g}}}$ be its Lie algebra. Let ${\displaystyle \mathrm {Ad} :G\rightarrow \mathrm {Aut} ({\mathfrak {g}})}$ denote the adjoint representation of ${\displaystyle G}$. Then the coadjoint representation ${\displaystyle K:G\rightarrow \mathrm {Aut} ({\mathfrak {g}}^{*})}$ is defined as ${\displaystyle \mathrm {Ad} ^{*}(g):=\mathrm {Ad} (g^{-1})^{*}}$. More explicitly,

${\displaystyle \langle K(g)F,Y\rangle =\langle F,\mathrm {Ad} (g^{-1})Y\rangle }$ for ${\displaystyle g\in G,Y\in {\mathfrak {g}},F\in {\mathfrak {g}}^{*},}$

where ${\displaystyle \langle F,Y\rangle }$ denotes the value of a linear functional ${\displaystyle F}$ on a vector ${\displaystyle Y}$.

Let ${\displaystyle K_{*}}$ denote the representation of the Lie algebra ${\displaystyle {\mathfrak {g}}}$ on ${\displaystyle {\mathfrak {g}}^{*}}$ induced by the coadjoint representation of the Lie group ${\displaystyle G}$. Then for ${\displaystyle X\in {\mathfrak {g}},K_{*}(X)=-\mathrm {ad} (X)^{*}}$ where ${\displaystyle \mathrm {ad} }$ is the adjoint representation of the Lie algebra ${\displaystyle {\mathfrak {g}}}$. One may make this observation from the infinitesimal version of the defining equation for ${\displaystyle K}$ above, which is as follows :

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

A coadjoint orbit ${\displaystyle \Omega :={\mathcal {O}}(F)}$ for ${\displaystyle F}$ in the dual space ${\displaystyle {\mathfrak {g}}^{*}}$ of ${\displaystyle {\mathfrak {g}}}$ may be defined either extrinsically, as the actual orbit ${\displaystyle K(G)(F)}$ inside ${\displaystyle {\mathfrak {g}}^{*}}$, or intrinsically as the homogeneous space ${\displaystyle G/\mathrm {Stab} (F)}$ where ${\displaystyle \mathrm {Stab} (F)}$ is the stabilizer of ${\displaystyle F}$ 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 ${\displaystyle {\mathfrak {g}}^{*}}$ and carry a natural symplectic structure. On each orbit ${\displaystyle \Omega }$, there is a closed non-degenerate ${\displaystyle G}$-invariant 2-form ${\displaystyle \sigma _{\Omega }}$ inherited from ${\displaystyle {\mathfrak {g}}}$ in the following manner. Let ${\displaystyle B_{F}}$ be an antisymmetric bilinear form on ${\displaystyle {\mathfrak {g}}}$ defined by,

${\displaystyle B_{F}(X,Y):=\langle F,[X,Y]\rangle ,X,Y\in {\mathfrak {g}}}$

Then one may define ${\displaystyle \sigma _{\Omega }\in \mathrm {Hom} (\Lambda ^{2}(\Omega ),\mathbb {R} )}$ by

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

The well-definedness, non-degeneracy, and ${\displaystyle G}$-invariance of ${\displaystyle \sigma _{\Omega }}$ follow from the following facts:

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

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

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

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

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