coadjoint orbit
Let G be a Lie group, and 𝔤 its Lie algebra. Then G has a natural action on 𝔤* called the coadjoint action, since it is dual to the adjoint action of G on 𝔤. The orbits of this action are submanifolds
of 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 λ∈𝔤* is typically denoted 𝒪λ.
The tangent space Tλ𝒪λ is naturally idenified by the action with 𝔤/𝔯λ, where 𝔯λ is the Lie algebra of the stabilizer of λ. The symplectic form on 𝒪λ is given by ωλ(X,Y)=λ([X,Y]). This is obviously anti-symmetric and non-degenerate since λ([X,Y])=0 for all Y∈𝔤 if and only if X∈𝔯λ. 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 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 |