Unitary highest weight modules for $\mathfrak{su}(p, q)$ and $\mathfrak{so}^{*}(2n)$ with fixed integral infinitesimal character

TL;DR

This paper classifies unitary highest weight modules with fixed integral infinitesimal character for certain Hermitian Lie algebras, covering both regular and singular cases, and identifies unitarizable modules within their highest weight structures.

Contribution

It provides a comprehensive classification of unitary highest weight modules for $rak{su}(p,q)$ and $rak{so}^*(2n)$, including both regular and singular cases, extending previous results.

Findings

Identified unitarizable modules within the highest weight orbit for $rak{su}(p,q)$.

Extended classification to singular cases for the considered Lie algebras.

Connected the results to earlier classifications for other Hermitian Lie algebras.

Abstract

We classify unitary highest weight modules with a given integral infinitesimal character for the real Lie algebras $\mathfrak{su}(p,q)$ and $\mathfrak{so}^*(2n)$. We treat both regular and singular cases. For $\mathfrak{su}(p,q)$ we identify the unitarizable modules in the Hasse diagrams of the highest weight orbit. Analogous results for the other Hermitian Lie algebras were given in our earlier publications.