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 $K : G \rightarrow \mathrm{Aut}(\mathfrak{g}^*)$ 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 $X, Y \in \mathfrak{g}, F \in \mathfrak{g}^*$. .

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 $F$ 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

$\sigma_{\Omega}(F)(K_{*}(X)(F), K_{*}(Y)(F)) := B_F(X, Y)$.

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 $G$ 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 at PlanetMath.org.