Invariant differential operator

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 R n {displaystyle mathbb {R} ^{n}} , functions on a manifold, vector valued functions, vector fields, or, more generally, sections of a vector bundle.The sphere (here shown as a red circle) as a conformal homogeneous manifold. 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 R n {displaystyle mathbb {R} ^{n}} , functions on a manifold, vector valued functions, vector fields, or, more generally, sections of a vector bundle. In an invariant differential operator D {displaystyle D} , the term differential operator indicates that the value D f {displaystyle Df} of the map depends only on f ( x ) {displaystyle f(x)} and the derivatives of f {displaystyle f} in x {displaystyle x} . The word invariant indicates that the operator contains some symmetry. This means that there is a group G {displaystyle 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. Let M = G/H be a homogeneous space for a Lie group G and a Lie subgroup H. Every representation ρ : H → A u t ( V ) {displaystyle ho :H ightarrow mathrm {Aut} (mathbb {V} )} gives rise to a vector bundle Sections φ ∈ Γ ( V ) {displaystyle varphi in Gamma (V)} 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 φ {displaystyle varphi } in Γ ( V ) {displaystyle 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.