Action integrals and infinitesimal characters

Abstract

Let $G$ be a reductive Lie group and $\mathcal{O}$ the coadjoint orbit of a hyperbolic element of ${\mathfrak{g}}^{\ast }$. 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 $\pi$.

Completely revised version to appear in Letters in Mathematical Physics