Let $(M,\omega)$ be a symplectic manifold, $G$ a Lie group acting on that manifold, $\fr g$ its Lie algebra, and $\fr g^*$ the dual of the Lie algebra. This action induces a map $\alpha:\fr g\to\fr X(M)$ where $\fr X(M)$ is the Lie algebra of vector fields on $M$ , such that $\exp(tX)(m)=\rho_t(m)$ where $\rho$ is the flow of $\alpha(X)$ . Then a moment map $\mu:M\to\fr g^*$ for the action of $G$ is a map such that

$$H_{\mu(X)}=\alpha(X).$$ Here $\mu(X)(m)=\mu(m)(X)$ , that is, $\mu(m)$ is a covector, so we apply it to the vector $X$ and get a scalar function $\mu(X)$ , and $H_{\mu(X)}$ is its Hamiltonian vector field.

Generally, the moment maps we are interested in are equivariant with respect to the coadjoint action, that is, they satisfy $$\Ad^*_g\circ\mu=\mu\circ g.$$