Lifting Hamiltonian loops to isotopies in fibrations

Let $G$ be a Lie group, $H$ a closed subgroup and $M$ the homogeneous space $G/H$. Each representation $\Psi$ 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 ${\rm Diff}(M)$, such that, each vector field $Z\in{\rm Lie}({\mathcal G})$ 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 $M_{\Delta}$ be the toric manifold determined by the Delzant polytope $\Delta$. We put $\varphi_{\bf b}$ for the the loop in the Hamiltonian group of $M_{\Delta}$ defined by the lattice vector ${\bf b}$. We give a sufficient condition, in terms of the mass center of $\Delta$, for the loops $\varphi_{\bf b}$ and $\varphi_{\bf\tilde b}$ to be homotopically inequivalent.