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 in 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.
Read more about this topic: Invariant Differential Operator
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