Thomas’s D-Calculus, Parabolic Invariant Theory, and Conformal Invariants

This chapter discusses the connections and the consequences of several pieces of recent work concerning conformally invariant differential operators. A conformal manifold is a smooth n-dimensional manifold M equipped with an equivalence class of metrics, the elements of which are related by conformal rescalings. Further, a vector bundle construction has been “discovered” independently by several people in the last few years, as a generalisation of the theory of “local twistors”. T. Y. Thomas first produced a version of it at about the same time as Cartan was developing his “conformal connection”, to which it is intimately related. Every invariant is a sum of invariants, each of which is a homogeneous polynomial of some degree d. Also, each invariant is uniquely a sum of an odd invariant and an even invariant, where “odd” and “even” refer to behaviour under reflections. Thus it suffices to consider invariants homogeneous of some degree and of definite parity.