A System of Third-Order Differential Operators Conformally Invariant under $\mathfrak{sl}(3,\mathbb{C})$ and $\mathfrak{so}(8,\mathbb{C})$

TL;DR

This paper constructs a new third-order conformally invariant differential operator system for specific Lie algebra types, resolving previously open cases and linking the construction to homomorphisms between generalized Verma modules.

Contribution

It explicitly constructs the missing third-order invariant systems for type A2 and D4 Lie algebras, expanding the understanding of conformally invariant differential operators.

Findings

Existence of third-order conformally invariant systems for A2 and D4.

Explicit construction of these systems.

Connection to homomorphisms between generalized Verma modules.

Abstract

In earlier work, Barchini, Kable, and Zierau constructed a number of conformally invariant systems of differential operators associated to Heisenberg parabolic subalgebras in simple Lie algebras. The construction was systematic, but the existence of such a system was left open in two cases, namely, the $\Omega_3$ system for type $A_2$ and type $D_4$. Here, such a system is shown to exist for both cases. The construction of the system may also be interpreted as giving an explicit homomorphism between generalized Verma modules.