Structure of the coadjoint orbits of Lie groups

Abstract

We study the geometrical structure of the coadjoint orbits of an arbitrary complex or real Lie algebra ${\mathfrak g}$ containing some ideal ${\mathfrak n}$. It is shown that any coadjoint orbit in ${\mathfrak g}^*$ is a bundle with the affine subspace of ${\mathfrak g}^*$ as its fibre. This fibre is an isotropic submanifold of the orbit and is defined only by the coadjoint representations of the Lie algebras ${\mathfrak g}$ and ${\mathfrak n}$ on the dual space ${\mathfrak n}^*$. The use of this fact and an application of methods of symplectic geometry give a new insight into the structure of coadjoint orbits and allow us to generalize results derived earlier in the case when ${\mathfrak g}$ is a split extension using the Abelian ideal ${\mathfrak n}$ (a semidirect product). As applications, a new proof of the formula for the index of Lie algebra and a necessary condition of integrality of a coadjoint orbit are obtained.