Positive Energy Representations, Holomorphic Discrete Series and Finite-Dimensional Irreps

TL;DR

This paper explores the relationships among positive energy, holomorphic discrete series, and finite-dimensional irreducible representations of semi-simple Lie groups, with detailed analysis of conformal groups SO(n,2) for n=1,3,4.

Contribution

It explicitly clarifies the connections between different types of representations of semi-simple Lie groups, especially for conformal groups, providing detailed case studies.

Findings

Explicit relations between positive energy and discrete series representations.

Detailed analysis of SO(n,2) for n=1,3,4.

Insights into the structure of finite-dimensional irreps.

Abstract

Let G be a semi-simple non-compact Lie group with unitary lowest/highest weight representations. We consider explicitly the relation between three types of representations of G: positive energy (unitary lowest weight)representations, (holomorphic) discrete series representations and non-unitary finite-dimensional irreps. We consider mainly the conformal groups SO(n,2) treating in full detail the cases n=1,3,4.