Semisimple coadjoint orbits and cotangent bundles

Abstract

Abstract. Semisimple (co)adjoint orbits through real hyperbolic elements are well-known to be symplectomorphic to cotangent bundles. We provide a new proof of this fact based on elementary results on both Lie theory and symplectic geometry. Our proof establishes a new connection between the Iwasawa horospherical projection and the symplectic geometry of real hyperbolic (co)adjoint orbits.

Canonical domains for coadjoint orbits

“…Remark The real hyperbolic orbit $$X^\vee$$ of the non‐compact semisimple Lie algebra $$\mathfrak {h}$$ is canonically symplectomorphic to the cotangent bundle over the Lagrangian $$L$$, which is its submanifold of real flags (the intersection with the ‐1‐eigen‐space in the Cartan decomposition) [19]: $$$$\begin{equation*} \Xi : (X^\vee , \Re \Omega )\rightarrow (T^*L,d\lambda ). \end{equation*}$$$$The Iwasawa ruling together with twice the Killing form identify the orbit with the cotangent bundle of the submanifold of real flags.…”

“…The diffeomorphism Remark 4.8. The real hyperbolic orbit 𝑋 ∨ of the non-compact semisimple Lie algebra 𝔥 is canonically symplectomorphic to the cotangent bundle over the Lagrangian 𝐿, which is its submanifold of real flags (the intersection with the -1-eigen-space in the Cartan decomposition) [19]:…”