Coadjoint orbit - Imported Knowlpedia Content

Definition

Let $G$ be a finite-dimensional Lie group with Lie algebra $\mathfrak g$ , and let $\lambda\in\mathfrak g^*$ . The coadjoint orbit through $\lambda$ is the orbit

\mathcal O_\lambda=G\cdot\lambda =\{\operatorname{Ad}_g^*\lambda:g\in G\}\subseteq\mathfrak g^*

for the coadjoint action. If $G_\lambda=\{g\in G:\operatorname{Ad}_g^*\lambda=\lambda\}$ is the stabilizer, then $\mathcal O_\lambda$ carries the canonical immersed-manifold structure for which $G/G_\lambda\to\mathcal O_\lambda$ is a $G$ -equivariant diffeomorphism. Its dimension is $\dim G-\dim G_\lambda$ .

Tangent space and homogeneous structure

At $\mu\in\mathcal O_\lambda$ , the tangent space is

T_\mu\mathcal O_\lambda =\{\operatorname{ad}_X^*\mu:X\in\mathfrak g\}.

Its kernel presentation identifies this tangent space with $\mathfrak g/\mathfrak g_\mu$ , where $\mathfrak g_\mu$ is the Lie algebra of the stabilizer. Thus each coadjoint orbit is a homogeneous space.

Kirillov–Kostant–Souriau form

Every coadjoint orbit has a canonical invariant symplectic form. With $X^\#_\mu=\frac{d}{dt}|_{0}\operatorname{Ad}_{\exp(tX)}^*\mu$ , one common convention is

\omega_\mu(X^\#_\mu,Y^\#_\mu)=\langle\mu,[X,Y]\rangle.

This is well defined, closed, and nondegenerate, so $\mathcal O_\lambda$ is a symplectic manifold. This structure underlies the orbit method described in Kirillov, “The Method of Orbits”.

Examples and sign conventions

For an abelian Lie group every coadjoint orbit is a single point. After identifying $\mathfrak{so}(3)^*\cong\mathbb R^3$ , the nonzero coadjoint orbits of $\mathrm{SO}(3)$ are spheres. Authors who define fundamental vector fields or moment maps with the opposite sign write the negative of the displayed symplectic form; the underlying orbit is unchanged.

References

A. A. Kirillov, Elements of the Theory of Representations, Springer, 1976. DOI record. Relevant: “The Method of Orbits.”
Ana Cannas da Silva, Lectures on Symplectic Geometry, Springer, 2008. DOI record. Relevant: Hamiltonian group actions and coadjoint orbits.