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

TL;DR

This paper constructs a new third-order conformally invariant differential operator system for the Lie algebra of type D4, resolving previously open cases and linking to homomorphisms between generalized Verma modules.

Contribution

It provides the first explicit construction of a conformally invariant third-order differential operator system in the remaining anomalous cases for type D4.

Findings

Existence of a conformally invariant third-order system for D4.

Explicit construction of the differential operator system.

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 several anomalous cases. Here, a conformally invariant system is shown to exist in the most interesting of these remaining cases. The construction may also be interpreted as giving an explicit homomorphism between generalized Verma modules for the Lie algebra of type $D_4$.