Let the Lie group $G$ act on the smooth manifold $X$ with the map $(g,x)\to gx$. In any point $x\in X$, the differential of this map induces a linear map: $$ \mu:T_e G \to T_xX\;, $$ and globally, if $\mathfrak{g}$ is the Lie algebra of $G$, a map $\mathfrak{g}\to \mathfrak{X}(X)$. (If I'm not mistaken, this should be even a Lie algebra homomorphism.) If $X$ is symplectic and $G$ acts by symplectomorphisms, this map is known as the moment map, because it gives the momenta of physics.
What is its name in general? And where can I find information about it?
Thanks.
EDIT (see Mike Miller's comment below):
Note that a linear map $\mathfrak{g}\to T_xX$ is (basically) the same as an element of $\mathfrak{g}^*\otimes T_xX$, which is (basically) the same as a map $T^*_xX\to \mathfrak{g}^*$.
In the case of symplectic manifolds, the map $m:T^*_xX\to \mathfrak{g}^*$ is composed with the symplectic gradient $gr:f\mapsto(df)^\sharp$, to give: $$ gr^*m: X\to T^*X\to \mathfrak{g}^*\;, $$ and this is called the moment map (Souriau, "Structure of Dynamical Systems"). But this works only for symplectic manifolds, because in general there is no canonical way to map functions $\mathfrak{g}\to \mathfrak{X}(X)$ into maps $X\to \mathfrak{g}^*$. The first one is just the differential of the group action, and so it is naturally defined in general.