= Solution
A <Hamiltonian group action> is a <smooth> <Lie group action> of $G$ on $(M,\omega)$ by <symplectomorphisms>, together with an equivariant <moment map> $\mu:M\to\mathfrak g^*$. For $\xi\in\mathfrak g$, let $\xi_M(p)=\left.\frac d{dt}\right|_0\exp(t\xi)\cdot p$ be its <fundamental vector field>. In our sign convention the defining identities are
$$
\boxed{d\langle\mu,\xi\rangle=\iota_{\xi_M}\omega,\qquad
\mu(g\cdot p)=\operatorname{Ad}_g^*\mu(p).}
$$
Here the left <coadjoint action> means $(\operatorname{Ad}_g^*\nu)(\xi)=\nu(\operatorname{Ad}_{g^{-1}}\xi)$. Each component $\mu^\xi$ is therefore a <Hamiltonian function> for the corresponding infinitesimal action. Equivariance is part of the definition; merely requiring each infinitesimal generator to be a <Hamiltonian vector field> is the weaker condition of a <weakly Hamiltonian action>. Reversing the defining sign of <Hamiltonian vector fields> reverses the <moment map> sign as well.
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