Proof of the Hamiltonian slice theorem

Question

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.

Answer 1

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

• Juan-Pablo Ortega and Tudor S. Ratiu: Momentum Maps and Hamiltonian Reduction
• Gerd Rudolph and Matthias Schmidt: Differential Geometry and Mathematical Physics

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.