  Let ${\cal O}$ be the orbit of $\eta\in{\frak g}^*$ under the coadjoint
action of the compact Lie group $G$. We give two formulae for calculating the
action integral along a closed Hamiltonian isotopy on ${\cal O}$. The first one
expresses this action in terms of a particular character of the isotropy
subgroup of $\eta$. In the second one is involved the character of an
irreducible representation of $G$.
