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Theorem(Guillemin Sternberg Marle) Let $(M, \omega, \mu) $ be a symplectic manifold together with a Hamiltonian group action. Let $p$ be a point in $M$ such that $O_p $ is contained in the zero level set of the moment map. Denote $G_p$ the stabilizer and $O_p$ the orbit of $p$. There is a $G$-equivariant symplectomorphism from a neighborhood of the zero section of the bundle $T^*G ×_{G_p} V_p$ equiped with a symplectic model to a neighborhood of the orbit $O_p$.

I'm looking for a reference of this theorem. ( The references that people recommend are the articles of Marle which is written in french and the article The normal form for the moment map by Guillemin and Sternberg which I can't download from internet). Could you please give some references where I can find the proof of this theorem.

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    $\begingroup$ I think it is covered in section 41, Symplectic slices and moment reconstruction, starting on p. 324 of Guillemin and Sternberg, Symplectic Techniques in Physics. $\endgroup$
    – Ben McKay
    Dec 9, 2021 at 13:24

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The following textbooks contain a proof of the symplectic slice theorem:

You might also have a look at my PhD thesis. One of the main results is a symplectic slice theorem in infinite dimensions. Thus there also a lot of analytical questions that need to be answered, so maybe not the best point to start learning this topic.

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  • $\begingroup$ .@Tobias Diez, thanks a lot for your helful answer! $\endgroup$
    – asma
    Dec 10, 2021 at 9:20

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