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 $gin Z(G)$. 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 $pi$.