Action Integrals and discrete series

TL;DR

This paper explores geometric interpretations of discrete series representations of complex semisimple Lie groups, linking constants associated with the representation to action integrals and providing bounds on fundamental groups of certain diffeomorphism subgroups.

Contribution

It introduces new geometric perspectives on discrete series representations, connecting representation constants to Hamiltonian dynamics and fundamental group bounds.

Findings

Constants are interpreted as action integrals of Hamiltonian loops.

Lower bounds for fundamental groups of diffeomorphism subgroups are established.

Values of the infinitesimal character are given geometric interpretations.

Abstract

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 $\pi$.