  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.
