Positive energy representations, holomorphic discrete series and 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 among 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 SOo(n, 2) treating in full detail the cases n = 1, 3, 4.