Let $(M,\omega)$ be a symplectic manifold endowed with a hamiltonian action of a torus $T$. Let $\mu : M \longrightarrow {Lie(T)}^*,$ be a moment map associated to this action. Let $S_M =\bigcap\limits_{m \in M} Stab(m)$, and $s_m$ be its lie algebra.
I have two questions about the moment map:
I) Why is the moment map constant on each connected component of $M^T$ ?
II) I know that for each $m \in M $, the image of tangent map of $\mu $ at m is $ Im(T_m \mu)={(s_m)}^\bot = \lbrace \eta \in Lie(T) \mid \langle \eta , X \rangle = 0 , \forall X \in s_m \rbrace $. How does this imply that the image of M by the moment map is an affine space directed by the linear space ${(s_m)}^\bot$ ?
I think it is helpful always to build intuition for Hamiltonian circle actions and then bring it over to the general case.
For a Hamiltonian circle actions there is a vector field $X$ (i.e. the integral curves of the vector field are the orbits of $S^1$-action) and smooth function $H : M \rightarrow \mathbb{R}$, subject to the equation: $$dH = \omega(X,\cdot) .$$
Now note that by the above identity a point $p$ is fixed by $S^{1}$ $\iff$ $X|_p=0$ $\iff$ $dH|_p=0$.
Hence the derivative of the Hamiltonian is exactly zero on components of the fixed point set and hence $H$ has to be constant on them. For higher dimensional torus actions picking a basis for the Lie algebra and applying this argument to each of the components give the argument.
For $2$ I think you should clarify since the image of the moment map is not always an affine space although it is a convex polytope (By AGS convexity theorem).