%0 Journal Article
%T Action Integrals and discrete series
%A Andr¨¦s ViŁża
%J Mathematics
%D 2011
%I arXiv
%X 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 $\pi$.
%U http://arxiv.org/abs/1108.1611v1