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 ${\mathfrak{g}}^{\ast }$ is a bundle with the affine subspace of ${\mathfrak{g}}^{\ast }$ 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 ${\mathfrak{n}}^{\ast }$. 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.