Heisenberg Parabolic Subgroup of SO*(8) and Invariant Differential Operators

Abstract

In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact algebra $so^*(8) \cong so(6,2)$. We use the maximal Heisenberg parabolic subalgebra ${\cal P} = {\cal M} \oplus {\cal A} \oplus {\cal N}$ with ${\cal M} = so^*(4)\oplus so(3)$. We give the main multiplets of indecomposable elementary representations (ERs). This includes the explicit parametrization of the invariant differential operators between the ERS. \\ Due to the recently established parabolic relations the multiplet classification results are valid also for the two algebras $so(p,q)$ (for $(p,q)=(5,3), (4,4)$) with maximal Heisenberg parabolic subalgebra: ${\cal P}' = {\cal M}' \oplus {\cal A}' \oplus {\cal N}'$, ${\cal M}' = so(p-2,q-2)\oplus sl(2,R)$, ${\cal M}'^C \cong {\cal M}^C.