# coadjoint orbit

Let $G$ be a Lie group, and $\U0001d524$ its Lie algebra^{}. Then $G$ has a natural action on ${\U0001d524}^{*}$ called the coadjoint action, since it is dual to the adjoint action of $G$ on $\U0001d524$. The orbits of this action are submanifolds^{} of ${\U0001d524}^{*}$ 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 $\lambda \in {\U0001d524}^{*}$ is typically denoted ${\mathcal{O}}_{\lambda}$.

The tangent space ${T}_{\lambda}{\mathcal{O}}_{\lambda}$ is naturally idenified by the action with $\U0001d524/{\U0001d52f}_{\lambda}$, where ${\U0001d52f}_{\lambda}$ 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 $Y\in \U0001d524$ if and only if $X\in {\U0001d52f}_{\lambda}$. 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 $\U0001d524$ 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 |