Conformally Invariant Operators via Curved Casimirs: Examples

TL;DR

This paper presents a method to construct conformally invariant differential operators using curved Casimir operators, applicable in various contexts and producing operators like powers of the Laplacian.

Contribution

It introduces a general scheme for building conformally invariant operators via curved Casimirs, demonstrating its effectiveness through explicit examples.

Findings

Method applies in regular and singular infinitesimal characters

Constructs both standard and non-standard operators

Includes examples with Laplacian powers

Abstract

We discuss a general scheme for a construction of linear conformally invariant differential operators from curved Casimir operators; we then explicitly carry this out for several examples. Apart from demonstrating the efficacy of the approach via curved Casimirs, this shows that this method applies both in regular and in singular infinitesimal character, and also that it can be used to construct standard as well as non--standard operators. The examples treated include conformally invariant operators with leading term, in one case, a square of the Laplacian, and in another case, a cube of the Laplacian.