coadjoint orbit

Let $G$ be a Lie group, and $\mathfrak{g}$ its Lie algebra. Then $G$ has a natural action on $\mathfrak{g}^{*}$ called the coadjoint action, since it is dual to the adjoint action of $G$ on $\mathfrak{g}$ . The orbits of this action are submanifolds of $\mathfrak{g}^{*}$ which carry a natural symplectic structure, and are in a certain sense, the minimal symplectic manifolds on which $G$ acts. The orbit through a point $\lambda\in\mathfrak{g}^{*}$ is typically denoted $\mathcal{O}_{\lambda}$ .

The tangent space $T_{\lambda}\mathcal{O}_{\lambda}$ is naturally idenified by the action with $\mathfrak{g}/\mathfrak{r}_{\lambda}$ , where $\mathfrak{r}_{\lambda}$ is the Lie algebra of the stabilizer of $\lambda$ . The symplectic form on $\mathcal{O}_{\lambda}$ is given by $\omega_{\lambda}(X,Y)=\lambda([X,Y])$ . This is obviously anti-symmetric and non-degenerate since $\lambda([X,Y])=0$ for all $Y\in\mathfrak{g}$ if and only if $X\in\mathfrak{r}_{\lambda}$ . This also shows that the form is well-defined.

There is a close association between coadoint orbits and the representation theory of $G$ , with irreducible representations being realized as the space of sections of line bundles on coadjoint orbits. For example, if $\mathfrak{g}$ is compact, coadjoint orbits are partial flag manifolds, and this follows from the Borel-Bott-Weil theorem.

Title	coadjoint orbit
Canonical name	CoadjointOrbit
Date of creation	2013-03-22 13:59:11
Last modified on	2013-03-22 13:59:11
Owner	bwebste (988)
Last modified by	bwebste (988)
Numerical id	5
Author	bwebste (988)
Entry type	Definition
Classification	msc 53D05