  Let $G$ be a complex semisimple Lie group and ${G}_{\mathbb R}$ a real form
that contains a compact Cartan subgroup $T_{\mathbb R}$. Let $\pi$ be a
discrete series representation of $G_{\mathbb R}$. We present geometric
interpretations in terms of concepts associated with the manifold
$M:=G_{\mathbb R}/T_{\mathbb R}$ of the constant $\pi(g)$, for $g\in
Z(G_{\mathbb R})$. For some relevant particular cases, we prove that this
constant is the action integral around a loop of Hamiltonian diffeomorphims of
$M$. As a consequence of these interpretations, we deduce lower bounds for the
cardinal of the fundamental group of some subgroups of ${\rm Diff}(M)$. We also
geometrically interpret the values of the infinitesimal character of the
differential representation of $\pi$.

Symplectic Geometry