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Equivariant geometry studies "manifolds" with an extra structure $G\times M\rightarrow M$.

Representation theory studies "Lie algebroids" with an extra structure $\Gamma(M,A)\times \Gamma(M,E)\rightarrow \Gamma(M,E)$.

There may be other similarities, but, I am not able to collect them together now.

Question is the following:

Is any relation between equivariant geometry (in appropriate sense) and representation theory (of appropriate geometric objects)?

I do not have any further data to support my guess. I will add more details if and when I find something interesting.

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  • $\begingroup$ I guess $G$ is a group and $M$ is a manifold in your definition of equivariant geometry, and so $M$ is probably also a manifold in your definition of representation theory. What are $A$ and $E$? $\endgroup$
    – LSpice
    Commented Jan 8, 2023 at 15:43
  • $\begingroup$ @LSpice sorry. I am not saying I know how to relate A and E with equivariant geometry. It is just a hunch.. $E\rightarrow M$ is a vector bundle, and $A\rightarrow M$ is a vector bundle with an extra structure on $\Gamma(M,A)$.. $\endgroup$
    – Praphulla Koushik
    Commented Jan 8, 2023 at 15:51
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    $\begingroup$ A manifold with an action of a Lie group gives rise to the associated Lie groupoid, which in its turn gives rise to the associated Lie algebroid. This gives a functor from manifolds with an action of a Lie group to Lie algebroids. Is there any other aspect of this correspondence that you are looking for? $\endgroup$
    – Dmitri Pavlov
    Commented Jan 8, 2023 at 18:29
  • $\begingroup$ @DmitriPavlov Every Lie group action on a manifold would give a Lie groupoid and that in turn gives a Lie algebroid.. That is not what I am referring to here.. I am thinking that the "data" of group action in equivariant set up looks similar to "data" of representation (for example of Lie algebroids).. I am trying to see if there is any relation between these two notions.. $\endgroup$
    – Praphulla Koushik
    Commented Jan 10, 2023 at 14:47

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