Invariant Differential Operators for Non-Compact Lie Groups: the Reduced SU(4,4) Multiplets

TL;DR

This paper systematically constructs invariant differential operators for non-compact Lie algebras, specifically focusing on the reduced multiplets of $su(n,n)$ for $n=4$, with implications for related algebras like $sl(8,\,\mathbb{R})$ and $su^*(8)$.

Contribution

It provides a complete classification of reduced multiplets containing physically relevant representations for $su(4,4)$, extending previous work to include minimal representations.

Findings

Constructed all reduced multiplets for $su(4,4)$ including minimal ones.

Extended results to $sl(8,\,\mathbb{R})$ and $su^*(8)$ via parabolic relations.

Identified physically relevant representations within these multiplets.

Abstract

In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact algebras $su(n,n)$. Earlier were given the main multiplets of indecomposable elementary representations for $n\leq 4$, and the reduced ones for $n=2,3$. Here we give all reduced multiplets containing physically relevant representations including the minimal ones for $n=4$. Due to the recently established parabolic relations the results are valid also for the algebras $sl(8,\mathbb{R})$ and $su^*(8)$ with suitably chosen maximal parabolic subalgebras.