coadjoint orbit

Let G be a Lie group, and 𝔤 its Lie algebraMathworldPlanetmath. 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 submanifoldsMathworldPlanetmath of 𝔤* which carry a natural symplectic structure, and are in a certain sense, the minimal symplectic manifoldsMathworldPlanetmath on which G acts. The orbit through a point λ𝔤* 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 stabilizerMathworldPlanetmath 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 Y𝔤 if and only if X𝔯λ. 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