Let $G$ be a compact Lie group and $H$ be a Lie subgroup of $G$. Suppose that $M$ is a smooth manifold on which $H$ acts from 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 $G \times_H M .$ Let $\iota: M \rightarrow G \times_H M$ be the map which associates to $ m \in M$ the class $[e,m]$.
Let $ m \in M$, my question is about the derivative of the map $\iota$ at $m$ $$ {d\iota} : T_mM \rightarrow T_{[e,m]} (G \times _H M ).$$
If $v=v_{\gamma(t)}\in T_mM$, what is $d\iota(v)$ ? (I have tried to apply the definition: $d\iota(v):= {\frac{d}{dt}}_{t=0} \iota(\gamma(t)) = {\frac{d}{dt}}_{t=0} [e,\gamma(t)]$, but I'm not sure what does this give, is it equal to [0,v] ?)