In mathematics and theoretical physics, an invariant differential operator is a kind of mathematical map from some objects to an object of similar type. These objects are typically functions on, functions on a manifold, vector valued functions, vector fields, or, more generally, sections of a vector bundle. In an invariant differential operator D, the term differential operator indicates that the value Df of the map depends only on f(x) and the derivatives of f in x. The word invariant indicates that the operator contains some symmetry. This means that there is a group G with a group action on the functions (or other objects in question) and this action is preserved by the operator: Usually, the action of the group has the meaning of a change of coordinates (change of observer) and the invariance means that the operator has the same expression in all admissible coordinates.

Invariance on homogeneous spaces

Let M = G/H be a homogeneous space for a Lie group G and a Lie subgroup H. Every representation gives rise to a vector bundle Sections can be identified with In this form the group G acts on sections via Now let V and W be two vector bundles over M. Then a differential operator that maps sections of V to sections of W is called invariant if for all sections \varphi in \Gamma(V) and elements g in G. All linear invariant differential operators on homogeneous parabolic geometries, i.e. when G is semi-simple and H is a parabolic subgroup, are given dually by homomorphisms of generalized Verma modules.

Invariance in terms of abstract indices

Given two connections \nabla and and a one form \omega, we have for some tensor. Given an equivalence class of connections [\nabla], we say that an operator is invariant if the form of the operator does not change when we change from one connection in the equivalence class to another. For example, if we consider the equivalence class of all torsion free connections, then the tensor Q is symmetric in its lower indices, i.e. . Therefore we can compute where brackets denote skew symmetrization. This shows the invariance of the exterior derivative when acting on one forms. Equivalence classes of connections arise naturally in differential geometry, for example:

Examples

Conformal invariance

Given a metric on, we can write the sphere S^{n} as the space of generators of the nil cone In this way, the flat model of conformal geometry is the sphere S^{n}=G/P with and P the stabilizer of a point in. A classification of all linear conformally invariant differential operators on the sphere is known (Eastwood and Rice, 1987).