Conformally Invariant Operators via Curved Casimirs: Examples

We discuss a 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 is general in that it 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. (Nevertheless the scheme discussed here does not recover all 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.