  Let $G$ be a reductive Lie group and ${\mathcal O}$ the coadjoint orbit of a
hyperbolic element of ${\frak g}^*$. By $\pi$ is denoted the unitary
irreducible representation of $G$ associated with ${\mathcal O}$ by the orbit
method. We give geometric interpretations in terms of concepts related to
${\mathcal O}$ of the constant $\pi(g)$, for $g\in Z(G)$.
  We also offer a description of the invariant $\pi(g)$ in terms of action
integrals and Berry phases. In the spirit of the orbit method we interpret
geometrically the infinitesimal character of the differential representation of
$\pi$.
