Nontrivial bundles of coadjoint orbits over $S^2$

Abstract

Let $G$ be a compact connected semisimple Lie group with Lie algebra $\mathfrak{g}$. Let $\mathcal{O}\subset\mathfrak{g}^*$ be a coadjoint orbit. The action of $G$ on $\mathcal{O}$ induces a morphism $\rho:G\to \mathrm{Homeo}(\mathcal{O})$. We prove that the induced map $\pi_1(\rho):\pi_1(G)\to\pi_1(\mathrm{Homeo}(\mathcal{O}))$ is injective. This strengthens a theorem of McDuff and Tolman (conjectured by Weinstein in 1989) according to which the analogous map $G\to\mathrm{Ham}(\mathcal{O})$ is injective on fundamental groups, where $\mathrm{Ham}(\mathcal{O})$ is the group of Hamiltonian diffeomorphisms of the standard symplectic structure on $\mathcal{O}$. To prove our theorem we associate to every nontrivial element of $\pi_1(G)$ a bundle over $S^2$ with fiber $\mathcal{O}$, using the standard patching construction. We then prove that the resulting bundle is topologically nontrivial by studying its cohomology. For this, we prove that it suffices to consider the case in which $G$ is simple and $\mathcal{O}$ is a minimal orbit, and then we prove our result for simple $G$ and minimal orbit $\mathcal{O}$ in a case by case analysis; the proof for the exceptional groups $E_6$ and $E_7$ relies on computer calculations, while all the other ones are addressed by hand. A basic tool in some of our computations is a generalization of Chevalley's formula to bundles of coadjoint orbits over $S^2$ that we prove in this paper.