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

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 $n\le 4$, 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 $s{u}^{\ast }(8)$ with suitably chosen maximal parabolic subalgebras.