Symplectic geometry of semisimple orbits

TL;DR

This paper demonstrates that certain coadjoint orbits of complex semisimple Lie groups are symplectomorphic to cotangent bundles of flag manifolds, extending the result to hyperbolic orbits in real semisimple Lie algebras.

Contribution

It establishes a symplectomorphism between specific coadjoint orbits and cotangent bundles, generalizing known results to hyperbolic orbits in real semisimple Lie algebras.

Findings

Coadjoint orbits with real eigenvalues are symplectomorphic to cotangent bundles of flag manifolds.

The result extends to hyperbolic orbits in real semisimple Lie algebras.

Provides a geometric understanding of the symplectic structure of these orbits.

Abstract

We prove that any coadjoint orbit with real eigenvalues of a complex semisimple Lie group, equipped with the real part of the canonical holomorphic symplectic form, is symplectomorphic to the cotangent bundle of a (partial) flag manifold. Moreover, we generalize this result to hyperbolic orbits in a real semisimple Lie algebra.