Linearization of hamiltonian torus action

Question

Let $(M,\omega)$ be a symplectic manifold with a hamiltonian effective torus action. Suppose it has an isolated fixed point $p$. Is it true that there exists an invariant neighborhood $U$ of $p$ such that the action on it is linear e.g. $U$ is symplectomorphic to an open ball in $\mathbb C^n$ with standard symplectic form and the torus action is linear? What can be said about local action around fixed submanifolds of dimension greater than $0$?

P.Pushkar told me that it must be true and I can read about it in the journal "Functional analysis and its applications" (Функциональный анализ и его применения) but unfortunately he didn't seem to manage to tell me the author, the name and the year of the publication. Perhaps you can find it.

Answer 1

There is always an equivariant local symplectomorphism with $T_pM$ with its 2-form and linear isotropy action, by the Moser-Weinstein proof. But that constant 2-form then has more possible “equivariant normal forms” than just $\sum dp_i\wedge dq_i$ — see e.g. Dellnitz-Melbourne (1993; pdf).

Answer 2

For fixed submanifolds, there is an analogous statement in Theorem 22.1 of V. Guillemin and S. Sternberg. Symplectic techniques in physics. Cambridge University Press, Cambridge, xi+468 pp, (1984).

This is called sometimes equivariant symplectic neighbourhood theorem or equivariant Darboux-Weinstein theorem.