Action integrals and infinitesimal characters

TL;DR

This paper explores the geometric and topological aspects of unitary representations of reductive Lie groups, linking action integrals, Berry phases, and infinitesimal characters to coadjoint orbits via the orbit method.

Contribution

It provides a geometric interpretation of the representation's character values and describes invariant quantities using action integrals and Berry phases.

Findings

Geometric interpretation of $ ext{tr}( ext{representation})$ for elements in the center.

Description of invariant $ ext{tr}( ext{representation})$ via action integrals.

Geometric understanding of the infinitesimal character of the differential representation.

Abstract

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 $g\in Z(G)$. 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 $\pi$.