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
, the term differential operator indicates that the value
The word invariant indicates that the operator contains some symmetry.
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.
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
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.
[1] Given an equivalence class of connections
, 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: Given a metric on
as the space of generators of the nil cone In this way, the flat model of conformal geometry is the sphere
A classification of all linear conformally invariant differential operators on the sphere is known (Eastwood and Rice, 1987).