Coadjoint representation

Coadjoint orbit

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

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

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

(i) The tangent space ${\displaystyle {\mathrm{T}}_{\nu }{\mathcal{O}}_{\mu }=\{-{\mathrm{a}\mathrm{d}}_X^{\ast }\nu :X\in \mathfrak{g}\}}$ may be identified with ${\displaystyle \mathfrak{g}/{\mathfrak{g}}_{\nu }}$, where ${\displaystyle {\mathfrak{g}}_{\nu }}$ is the Lie algebra of ${\displaystyle G_{\nu }}$.

(ii) The kernel of the map ${\displaystyle X\mapsto ⟨\nu ,[X,\cdot ]⟩}$ is exactly ${\displaystyle {\mathfrak{g}}_{\nu }}$.

(iii) The bilinear form ${\displaystyle ⟨\nu ,[\cdot ,\cdot ]⟩}$ on ${\displaystyle \mathfrak{g}}$ is invariant under ${\displaystyle G_{\nu }}$.

${\displaystyle \omega }$ is also closed. The canonical 2-form ${\displaystyle \omega }$ 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 ${\displaystyle ({\mathcal{O}}_{\mu },\omega )}$ is a Hamiltonian ${\displaystyle G}$-action with momentum map given by the inclusion ${\displaystyle {\mathcal{O}}_{\mu }\hookrightarrow {\mathfrak{g}}^{\ast }}$.

Examples