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:
D ( g ⋅ f ) = g ⋅ ( D f ) . {\displaystyle D(g\cdot f)=g\cdot (Df).}
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 ρ : H → A u t ( V ) {\displaystyle \rho :H\rightarrow \mathrm {Aut} (\mathbb {V} )} gives rise to a vector bundle
V = G × H V where ( g h , v ) ∼ ( g , ρ ( h ) v ) ∀ g ∈ G , h ∈ H and v ∈ V . {\displaystyle V=G\times _{H}\mathbb {V} \;{\text{where}}\;(gh,v)\sim (g,\rho (h)v)\;\forall \;g\in G,\;h\in H\;{\text{and}}\;v\in \mathbb {V} .}
Sections φ ∈ Γ ( V ) {\displaystyle \varphi \in \Gamma (V)} can be identified with
Γ ( V ) = { φ : G → V : φ ( g h ) = ρ ( h − 1 ) φ ( g ) ∀ g ∈ G , h ∈ H } . {\displaystyle \Gamma (V)=\{\varphi :G\rightarrow \mathbb {V} \;:\;\varphi (gh)=\rho (h^{-1})\varphi (g)\;\forall \;g\in G,\;h\in H\}.}
In this form the group G acts on sections via
( ℓ g φ ) ( g ′ ) = φ ( g − 1 g ′ ) . {\displaystyle (\ell _{g}\varphi )(g')=\varphi (g^{-1}g').}
Now let V and W be two vector bundles over M. Then a differential operator
d : Γ ( V ) → Γ ( W ) {\displaystyle d:\Gamma (V)\rightarrow \Gamma (W)}
that maps sections of V to sections of W is called invariant if
d ( ℓ g φ ) = ℓ g ( d φ ) . {\displaystyle d(\ell _{g}\varphi )=\ell _{g}(d\varphi ).}
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.
Invariance in terms of abstract indices Given two connections ∇ {\displaystyle \nabla } and ∇ ^ {\displaystyle {\hat {\nabla }}} and a one form ω {\displaystyle \omega } , we have
∇ a ω b = ∇ ^ a ω b − Q a b
c ω c {\displaystyle \nabla _{a}\omega _{b}={\hat {\nabla }}_{a}\omega _{b}-Q_{ab}{}^{c}\omega _{c}}
for some tensor Q a b
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