On the classification of unitary highest weight modules in the exceptional cases

TL;DR

This paper completes the classification of unitary highest weight modules for certain exceptional Lie groups, building on previous work and providing detailed descriptions of modules with specific infinitesimal characters.

Contribution

It extends the classification of unitary highest weight modules to the remaining exceptional Lie groups, using the Dirac inequality and PRV product methods.

Findings

Complete classification for $SO_{e}(2, n)$, $E_{6(-14)}$, and $E_{7(-25)}$.

Description of modules with given infinitesimal characters.

Methodological extension of previous classification techniques.

Abstract

In our previous paper, we gave a complete classification of the unitary highest weight modules for the universal covers of the Lie groups $Sp(2n, \mathbb{R}), SO^{*}(2n)$ and $SU(p, q)$, using the Dirac inequality and the so called PRV product. In this paper, we complete the classification of the unitary highest weight modules for the remaining cases; i.e., universal covers of the Lie groups $SO_{e}(2, n)$, $E_{6(-14)}$ and $E_{7(-25)}$. We also describe unitary highest weight modules with given infinitesimal characters.