Equivariant symplectomorphisms preserve moment maps up to a coadjoint-fixed covector

For every $\xi \in \mathfrak{g}$ the fundamental field is $\varphi$-related to itself: differentiating the identity $\varphi ({\exp }_G(-t\xi )\cdot p)={\exp }_G(-t\xi )\cdot \varphi (p)$, which holds because $\varphi$ commutes with the action, gives $d{\varphi }_p({\xi }_M(p))={\xi }_M(\varphi (p))$.

The composite $\mu \circ \varphi$ is coadjoint equivariant: $(\mu \circ \varphi )(g\cdot p)=\mu (\varphi (g\cdot p))=\mu (g\cdot \varphi (p))=g\cdot \mu (\varphi (p))=g\cdot (\mu \circ \varphi )(p)$ for all $g,p$.

The composite satisfies the component moment equations. Indeed, for $v\in T_{p}M$, $d(\mu \circ \varphi {)}_p^{\xi }(v)=d{\mu }_{\varphi (p)}^{\xi }(d{\varphi }_{p}v)=-{\omega }_{\varphi (p)}({\xi }_M(\varphi (p)),d{\varphi }_{p}v)=-{\omega }_{\varphi (p)}(d{\varphi }_p{\xi }_M(p),d{\varphi }_{p}v)=-{\omega }_p({\xi }_M(p),v),$ where step 1.1 identifies the fundamental field at $\varphi (p)$ with $d{\varphi }_p{\xi }_M(p)$ and symplecticity of $\varphi$ removes the differential.

By steps 1.2 and 2.1 the composite $\mu \circ \varphi$ is an equivariant moment map for the same action as $\mu$. Both are equivariant moment maps on the connected manifold $M$, so [F4] provides $\delta \in ({\mathfrak{g}}^{\ast }{)}^G$ with $\mu \circ \varphi -\mu =\delta$.

The difference vanishes exactly when $\delta =0$. Adding any constant covector $c$ to the moment map does not change this difference, because $(\mu +c)\circ \varphi -(\mu +c)=\mu \circ \varphi -\mu$. If $p$ is a fixed point of $\varphi$, however, then evaluating step 3.1 at $p$ gives $\delta =\mu (\varphi (p))-\mu (p)=0$, so $\mu \circ \varphi =\mu$.