Lifting Hamiltonian loops to isotopies in fibrations

Let $G$ be a Lie group, $H$ a closed subgroup and $M$ the homogeneous space $G\mathrm{/}H$. Each representation $\mathrm{\Psi }$ of $H$ determines a $G$-equivariant principal bundle $\mathcal{P}$ on $M$ endowed with a $G$-invariant connection. We consider subgroups $\mathcal{G}$ of the diffeomorphism group $\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}(M)$, such that, each vector field $Z\in \mathrm{L}\mathrm{i}\mathrm{e}(\mathcal{G})$ admits a lift to a preserving connection vector field on $\mathcal{P}$. We prove that #\,\pi_1({\mathcal G})\geq #\,\Psi(Z(G)). This relation is applicable to subgroups $\mathcal{G}$ of the Hamiltonian groups of the flag varieties of a semisimple group $G$. Let $M_{\mathrm{\Delta }}$ be the toric manifold determined by the Delzant polytope $\mathrm{\Delta }$. We put ${\phi }_{\mathbf{b}}$ for the the loop in the Hamiltonian group of $M_{\mathrm{\Delta }}$ defined by the lattice vector $\mathbf{b}$. We give a sufficient condition, in terms of the mass center of $\mathrm{\Delta }$, for the loops $\varphi_{\bf b}$and$\varphi_{\bf\tilde b}$ to be homotopically inequivalent.