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 orbits. 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 $\mathrm{A}\mathrm{d}:G\to \mathrm{A}\mathrm{u}\mathrm{t}(\mathfrak{𝔤})$ denote the adjoint representation of $G$. Then the coadjoint representation ${\mathrm{A}\mathrm{d}}^{*}:G\to \mathrm{G}\mathrm{L}({\mathfrak{𝔤}}^{*})$ is defined by

$⟨{\mathrm{A}\mathrm{d}}_g^{*}\mu ,Y⟩=⟨\mu ,{\mathrm{A}\mathrm{d}}_g^{-1}Y⟩=⟨\mu ,{\mathrm{A}\mathrm{d}}_{g^{-1}}Y⟩$ for $g\in G,Y\in \mathfrak{𝔤},\mu \in {\mathfrak{𝔤}}^{*},$

where $⟨\mu ,Y⟩$ denotes the value of the linear functional $\mu$ on the vector $Y$. Let ${\mathrm{a}\mathrm{d}}^{*}$ denote the representation of the Lie algebra $\mathfrak{𝔤}$ on ${\mathfrak{𝔤}}^{*}$ induced by the coadjoint representation of the Lie group $G$. Then the infinitesimal version of the defining equation for ${\mathrm{A}\mathrm{d}}^{*}$ reads:

$⟨{\mathrm{a}\mathrm{d}}_X^{*}\mu ,Y⟩=⟨\mu ,-{\mathrm{a}\mathrm{d}}_{X}Y⟩=-⟨\mu ,[X,Y]⟩$ for $X,Y\in \mathfrak{𝔤},\mu \in {\mathfrak{𝔤}}^{*}$

where $\mathrm{a}\mathrm{d}$ is the adjoint representation of the Lie algebra $\mathfrak{𝔤}$.

Coadjoint orbit

A coadjoint orbit ${{𝒪}}_{\mu }$ for $\mu$ in the dual space ${\mathfrak{𝔤}}^{*}$ of $\mathfrak{𝔤}$ may be defined either extrinsically, as the actual orbit ${\mathrm{A}\mathrm{d}}_G^{*}\mu$ inside ${\mathfrak{𝔤}}^{*}$, or intrinsically as the homogeneous space $G/G_{\mu }$ where $G_{\mu }$ is the stabilizer of $\mu$ 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 ${{𝒪}}_{\mu }$, there is a closed non-degenerate $G$-invariant 2-form $\omega \in {\mathrm{\Omega }}^2({{𝒪}}_{\mu })$ inherited from $\mathfrak{𝔤}$ in the following manner:

${\omega }_{\nu }({\mathrm{a}\mathrm{d}}_X^{*}\nu ,{\mathrm{a}\mathrm{d}}_Y^{*}\nu ):=⟨\nu ,[X,Y]⟩,\nu \in {{𝒪}}_{\mu },X,Y\in \mathfrak{𝔤}$.

The well-definedness, non-degeneracy, and $G$-invariance of $\omega$ follow from the following facts: (i) The tangent space ${\mathrm{T}}_{\nu }{{𝒪}}_{\mu }=\{-{\mathrm{a}\mathrm{d}}_X^{*}\nu :X\in \mathfrak{𝔤}\}$ may be identified with $\mathfrak{𝔤}/{\mathfrak{𝔤}}_{\nu }$, where ${\mathfrak{𝔤}}_{\nu }$ is the Lie algebra of $G_{\nu }$. (ii) The kernel of the map $X\mapsto ⟨\nu ,[X,\cdot ]⟩$ is exactly ${\mathfrak{𝔤}}_{\nu }$. (iii) The bilinear form $⟨\nu ,[\cdot ,\cdot ]⟩$ on $\mathfrak{𝔤}$ is invariant under $G_{\nu }$. $\omega$ is also closed. The canonical 2-form $\omega$ is sometimes referred to as the Kirillov-Kostant-Souriau symplectic form or KKS form on the coadjoint orbit.

Properties of coadjoint orbits

The coadjoint action on a coadjoint orbit $({{𝒪}}_{\mu },\omega )$ is a Hamiltonian $G$-action with momentum map given by the inclusion ${{𝒪}}_{\mu }\hookrightarrow {\mathfrak{𝔤}}^{*}$.

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 orbit". PlanetMath.