Invariant differential operator

<h2 id="invariance-on-homogeneous-spaces">Invariance on homogeneous spaces</h2>
Let M = G/H be a <a href="/facts/Homogeneous_space/YGebqcJA">homogeneous space</a> for a <a href="/facts/Lie_group/a9jCQyjF">Lie group</a> G and a Lie subgroup H. Every <a href="/facts/Representation_(mathematics)/mpsX51I9">representation</a> 
 
 
 
 ρ
 :
 H
 →
 
 A
 u
 t
 
 (
 
 V
 
 )
 
 
 {\displaystyle \rho :H\rightarrow \mathrm {Aut} (\mathbb {V} )}
 
 gives rise to a <a href="/facts/Vector_bundle/cXACAa1I">vector bundle</a>

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 <a href="/facts/Vector_bundle/cXACAa1I">vector bundles</a> 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 <a href="/facts/Parabolic_geometry_(differential_geometry)/x1rhxLFX">parabolic geometries</a>, i.e. when G is semi-simple and H is a <a href="/facts/Borel_subgroup/F80yLMqA">parabolic subgroup</a>, are given dually by homomorphisms of <a href="/facts/Generalized_Verma_module/I4np80qA">generalized Verma modules</a>.

<h2 id="invariance-in-terms-of-abstract-indices">Invariance in terms of abstract indices</h2>
Given two <a href="/facts/Connection_(mathematics)/KURsCmIu">connections</a> 
 
 
 
 ∇
 
 
 {\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

c
 
 
 
 
 {\displaystyle Q_{ab}{}^{c}}
 
.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a> Given an <a href="/facts/Equivalence_class/1d2sh0TB">equivalence class</a> of connections 
 
 
 
 [
 ∇
 ]
 
 
 {\displaystyle [\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 <a href="/facts/Torsion_tensor/LVfDhPnW">torsion free</a> connections, then the tensor Q is symmetric in its lower indices, i.e. 
 
 
 
 
 Q
 
 a
 b

c
          
        
        =
        
          Q
          
            (
            a
            b
            )

c
 
 
 
 
 {\displaystyle Q_{ab}{}^{c}=Q_{(ab)}{}^{c}}
 
. Therefore we can compute

∇
          
            [
            a
          
        
        
          ω
          
            b
            ]
          
        
        =
        
          
            
              
                ∇
                ^
              
            
          
          
            [
            a
          
        
        
          ω
          
            b
            ]
          
        
        ,
      
    
    {\displaystyle \nabla _{[a}\omega _{b]}={\hat {\nabla }}_{[a}\omega _{b]},}

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:

<ul><li>in <a href="/facts/Conformal_geometry/tuU181cE">conformal geometry</a> an equivalence class of connections is given by the Levi Civita connections of all <a href="/facts/Metric_(mathematics)/RkUptSo8">metrics</a> in the conformal class;</li>
<li>in <a href="/facts/Projective_geometry/rHKSaEbx">projective geometry</a> an equivalence class of connection is given by all connections that have the same <a href="/facts/Geodesics/fbWR6l80">geodesics</a>;</li>
<li>in <a href="/facts/CR_geometry/ECXTscxj">CR geometry</a> an equivalence class of connections is given by the Tanaka-Webster connections for each choice of pseudohermitian structure</li></ul>
<h2 id="examples">Examples</h2>
<ol><li>The usual <a href="/facts/Gradient/TsR7uukj">gradient</a> operator 
 
 
 
 ∇
 
 
 {\displaystyle \nabla }
 
 acting on real valued functions on <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a> is invariant with respect to all <a href="/facts/Euclidean_transformation/0KzWYQXl">Euclidean transformations</a>.</li>
<li>The <a href="/facts/Exterior_derivative/p7HA0wze">differential</a> acting on functions on a manifold with values in <a href="/facts/One-form/NQLbWmtD">1-forms</a> (its expression is      
 
 
 
 d
 =
 
 ∑
 
 j
 
 
 
 ∂
 
 j
 
 
 
 d
 
 x
 
 j
 
 
 
 
 {\displaystyle d=\sum _{j}\partial _{j}\,dx_{j}}
 
 in any local coordinates) is invariant with respect to all smooth transformations of the manifold (the action of the transformation on <a href="/facts/Differential_form/T9gyHtMv">differential forms</a> is just the <a href="/facts/Pullback_(differential_geometry)/EzMDBYfK">pullback</a>).</li>
<li>More generally, the <a href="/facts/Exterior_derivative/p7HA0wze">exterior derivative</a>      
 
 
 
 d
 :
 
 Ω
 
 n
 
 
 (
 M
 )
 →
 
 Ω
 
 n
 +
 1
 
 
 (
 M
 )
 
 
 {\displaystyle d:\Omega ^{n}(M)\rightarrow \Omega ^{n+1}(M)}
 
 that acts on n-forms of any smooth manifold M is invariant with respect to all smooth transformations. It can be shown that the exterior derivative is the only linear invariant differential operator between those bundles.</li>
<li>The <a href="/facts/Dirac_operator/1kdGnoTt">Dirac operator</a> in physics is invariant with respect to the <a href="/facts/Poincar%C3%A9_group/3IerRYG1">Poincaré group</a> (if we choose the proper <a href="/facts/Group_action_(mathematics)/Vh9VtUUw">action</a> of the <a href="/facts/Poincar%C3%A9_group/3IerRYG1">Poincaré group</a> on <a href="/facts/Spinor/sKFNnEwP">spinor</a> valued functions. This is, however, a subtle question and if we want to make this mathematically rigorous, we should say that it is invariant with respect to a group which is a <a href="/facts/Double_covering_group/IrRjtTL2">double cover</a> of the Poincaré group)</li>
<li>The <a href="/facts/Conformal_Killing_equation/guRXrHG0">conformal Killing equation</a>      
 
 
 
 
 X
 
 a
 
 
 ↦
 
 ∇
 
 (
 a
 
 
 
 X
 
 b
 )
 
 
 −
 
 
 1
 n
 
 
 
 ∇
 
 c
 
 
 
 X
 
 c
 
 
 
 g
 
 a
 b
 
 
 
 
 {\displaystyle X^{a}\mapsto \nabla _{(a}X_{b)}-{\frac {1}{n}}\nabla _{c}X^{c}g_{ab}}
 
 is a conformally invariant linear differential operator between vector fields and symmetric trace-free tensors.</li></ol>
<h2 id="conformal-invariance">Conformal invariance</h2>

Given a metric

g
        (
        x
        ,
        y
        )
        =
        
          x
          
            1
          
        
        
          y
          
            n
            +
            2
          
        
        +
        
          x
          
            n
            +
            2
          
        
        
          y
          
            1
          
        
        +
        
          ∑
          
            i
            =
            2
          
          
            n
            +
            1
          
        
        
          x
          
            i
          
        
        
          y
          
            i
          
        
      
    
    {\displaystyle g(x,y)=x_{1}y_{n+2}+x_{n+2}y_{1}+\sum _{i=2}^{n+1}x_{i}y_{i}}

on 
 
 
 
 
 
 R
 
 
 n
 +
 2
 
 
 
 
 {\displaystyle \mathbb {R} ^{n+2}}
 
, we can write the <a href="/facts/Sphere/jywYLNM2">sphere</a> 
 
 
 
 
 S
 
 n
 
 
 
 
 {\displaystyle S^{n}}
 
 as the space of generators of the <a href="/facts/Null_cone/sjkdQtdS">nil cone</a>

S
          
            n
          
        
        =
        {
        [
        x
        ]
        ∈
        
          
            R
            P
          
          
            n
            +
            1
          
        
        
        :
        
        g
        (
        x
        ,
        x
        )
        =
        0
        }
        .
      
    
    {\displaystyle S^{n}=\{[x]\in \mathbb {RP} _{n+1}\;:\;g(x,x)=0\}.}

In this way, the flat model of <a href="/facts/Conformal_geometry/tuU181cE">conformal geometry</a> is the sphere 
 
 
 
 
 S
 
 n
 
 
 =
 G
 
 /
 
 P
 
 
 {\displaystyle S^{n}=G/P}
 
 with 
 
 
 
 G
 =
 S
 
 O
 
 0
 
 
 (
 n
 +
 1
 ,
 1
 )
 
 
 {\displaystyle G=SO_{0}(n+1,1)}
 
 and P the stabilizer of a point in 
 
 
 
 
 
 R
 
 
 n
 +
 2
 
 
 
 
 {\displaystyle \mathbb {R} ^{n+2}}
 
. A classification of all linear conformally invariant differential operators on the sphere is known (Eastwood and Rice, 1987).<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Differential_operator/Nr15J0IE">Differential operators</a></li>
<li><a href="/facts/Laplace_invariant/JQ3blrF9">Laplace invariant</a></li>
<li><a href="/facts/Invariant_factorization_of_LPDOs/3PRDG7o4">Invariant factorization of LPDOs</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li>Slovák, Jan (1993). <a href="ftp://www.math.muni.cz/pub/math/people/Slovak/papers/vienna.ps">Invariant Operators on Conformal Manifolds</a>. Research Lecture Notes, University of Vienna (Dissertation).</li>
<li>Kolář, Ivan; Michor, Peter; Slovák, Jan (1993). <a href="https://web.archive.org/web/20170330154524/http://www.emis.de/monographs/KSM/kmsbookh.pdf">Natural operators in differential geometry</a> (PDF). Springer-Verlag, Berlin, Heidelberg, New York. Archived from <a href="http://www.emis.de/monographs/KSM/kmsbookh.pdf">the original</a> (PDF) on 2017-03-30. Retrieved 2011-01-05.</li>
<li>Eastwood, M. G.; Rice, J. W. (1987). <a href="http://projecteuclid.org/euclid.cmp/1104116840">"Conformally invariant differential operators on Minkowski space and their curved analogues"</a>. Commun. Math. Phys. 109 (2): 207–228. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1987CMaPh.109..207E">1987CMaPh.109..207E</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01215221">10.1007/BF01215221</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:121161256">121161256</a>.</li>
<li>Kroeske, Jens (2008). "Invariant bilinear differential pairings on parabolic geometries". PhD Thesis from the University of Adelaide. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/0904.3311">0904.3311</a>. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2009PhDT.......274K">2009PhDT.......274K</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Penrose and Rindler (1987). Spinors and Space Time. Cambridge Monographs on Mathematical Physics. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">M.G. Eastwood and J.W. Rice (1987). "Conformally invariant differential operators on Minkowski space and their curved analogues". Commun. Math. Phys. 109 (2): 207–228. Bibcode:1987CMaPh.109..207E. doi:10.1007/BF01215221. S2CID 121161256. <a href="http://projecteuclid.org/euclid.cmp/1104116840" target="_blank">http://projecteuclid.org/euclid.cmp/1104116840</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Dobrev, Vladimir (1988). "Canonical construction of intertwining differential operators associated with representations of real semisimple Lie groups". Rep. Math. Phys. 25 (2): 159–181. Bibcode:1988RpMP...25..159D. doi:10.1016/0034-4877(88)90050-X. <a href="http://www.journals.elsevier.com/reports-on-mathematical-physics/" target="_blank">http://www.journals.elsevier.com/reports-on-mathematical-physics/</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
</ol>

Invariant differential operator open-in-new

Invariant differential operator