Dirac inequality for highest weight Harish-Chandra modules I

TL;DR

This paper explores a direct approach to classifying unitary highest weight representations of Hermitian type Lie groups using the Dirac inequality, simplifying previous methods based on dual pairs and Jantzen's formula.

Contribution

It introduces a new application of the Dirac inequality to classify unitary highest weight modules more straightforwardly than prior techniques.

Findings

Demonstrates the effectiveness of Dirac inequality in classification

Provides a simplified proof of the unitary highest weight representation classification

Enhances understanding of representation theory for Hermitian Lie groups

Abstract

Let $G$ be a connected simply connected noncompact classical simple Lie group of Hermitian type. Then $G$ has unitary highest weight representations. The proof of the classification of unitary highest weight representations of $G$ given by Enright, Howe and Wallach is based on the Dirac inequality of Parthasarathy, Jantzen's formula and Howe's theory of dual pairs where one group in the pair is compact. In this paper we focus on the Dirac inequality which can be used to prove the classification in a more direct way.