Invariant Differential Pairings

Invariant differential operators appear often in mathematics and theoretical physics. The word invariant indicates that the operator in question contains some symmetry. The discovery and exploitation of these symmetries form an important part of research in these areas. Physical theories often require that the equations by which they are defined exhibit natural symmetries. This work is concerned with the extension of the study of invariant linear differential operators to bilinear differential pairings. This analysis is carried out for a wide class of geometric settings called parabolic geometries, examples of which are given by conformal geometry, CR geometry and projective geometry. We introduce the concept with a combination of differential geometric methods and representation theory. Various techniques such as jet bundles and tractor calculus are carefully explained. Supported by many concrete examples our exhibition is aimed at both mathematicians and physicists with an interest in the geometric analysis of partial differential equations.