Action integrals and infinitesimal characters

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 $\pi$.