Definition: Let $M$ be a smooth manifold on which acts a lie group $G$. Let $X$ be an element in the lie algebra $\mathfrak{g}$ of $G$, we associate to it the vector field $X_M$ called the fundamental vector field defined by $$X_M(m)= \frac{d}{dt}\biggr\vert_{t=0} e^{-tX}.m.$$
Let $G$ be a Lie group and let $H$ be a Lie subgroup of $H$. Let $M$ be a smooth manifold on which $H$ acts on the left.
Let's consider the action of $H$ on $G \times M$ : $$h((g,m)):= (gh,h^{-1}m), \quad h \in H , g \in G , m \in M, $$ and define the manifold $Z$ to be the quotient $G \times_H M .$
The group $G$ acts on $Z:= G \times _H M$ by $g'.[g,m]:= [g'g,m].$
Let $X \in \mathfrak{g}$, What is $X_Z$ ? i.e if $[g,m] \in G \times_H M$, then how can we continue this calculation:
$$X_Z([g,m])= \frac{d}{dt}\biggr\vert_{t=0} e^{-tX}.[g,m]= \frac{d}{dt}\biggr\vert_{t=0} [e^{-tX}g,m] ?$$
Any help would be very appreciated.