# Proof of the Hamiltonian slice theorem

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.

• 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. Dec 9, 2021 at 13:24