Kalkman element

It seems that there is a general theme which plays a role here. It is basically Kalkman map between Cartan and Weyl models of equivariant cohomology.

Restriction of form to orbit

Slight generalization

Generalization

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Restriction of form to orbit

Slight generalization

After the first step (which was ), we could have (instead of putting ) integrated over a submanifold where is some compact submanifold of . (Choice of a point is a particular case, when is one point.) If we take the whole group of diffeomorphisms of , then this prescription cooks out of an inhomogeneous differential form on the space of submanifolds .

Generalization

We have some representation of . Then Eq. (2) still holds, where should be compatible with the differential of .

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