Let be a Lie group, and its Lie algebra. Then has a natural action on called the coadjoint action, since it is dual to the adjoint action of on . The orbits of this action are submanifolds of which carry a natural symplectic structure, and are in a certain sense, the minimal symplectic manifolds on which acts. The orbit through a point is typically denoted .
The tangent space is naturally idenified by the action with , where is the Lie algebra of the stabilizer of . The symplectic form on is given by . This is obviously anti-symmetric and non-degenerate since for all if and only if . This also shows that the form is well-defined.
There is a close association between coadoint orbits and the representation theory of , with irreducible representations being realized as the space of sections of line bundles on coadjoint orbits. For example, if is compact, coadjoint orbits are partial flag manifolds, and this follows from the Borel-Bott-Weil theorem.
|Date of creation||2013-03-22 13:59:11|
|Last modified on||2013-03-22 13:59:11|
|Last modified by||bwebste (988)|