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 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,

$\langle K(g)F,Y\rangle =\langle F,\mathrm {Ad} (g^{-1})Y\rangle$ 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 $K_{*}=-\mathrm {ad} ^{*}$ 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 $X,Y\in {\mathfrak {g}},F\in {\mathfrak {g}}^{*}$. .

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 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 $\sigma _{\Omega }\in \Lambda ^{2}(\Omega )$ 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 $\mathrm {Stab} (F)$.

(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 on the coadjoint orbit.

Properties of coadjoint orbits

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

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, ISBN 0821835300, ISBN 978-0821835302

External links

• "Coadjoint representation". PlanetMath.