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Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$ and $H$ be a Lie subgroup of $G$ with Lie algebra $\mathfrak{h}$. Consider the action of $H$ on $G$ by right multiplication and the action of $H$ on $\mathfrak{g}/\mathfrak{h}$ induced from the adjoint action of $H$ on $\mathfrak{g}$ and we define the quotient space $ M:=G \times_H \mathfrak{g}/\mathfrak{h}$.

The manifold $G \times _H \mathfrak{g}/\mathfrak{h}$ (which is isomorphic to the tangent bundle $T(G/H))$ is a vector bundle over $G/H$, then for every $[g,X] \in M$, the tangent space $T_{[g,X]}M$ is isomorphic to $T_{[g]}(G/H) \times T_{[X]}(\mathfrak{g}/\mathfrak{h})$ and then is isomorphic to $\mathfrak{g}/\mathfrak{h} \times \mathfrak{g}/\mathfrak{h}$.

My question is how to construct an explicit isomorphism between the tangent space $T_{[g,X]} (G \times _H \mathfrak{g}/\mathfrak{h})$ and $\mathfrak{g}/\mathfrak{h} \times \mathfrak{g}/\mathfrak{h}$, it means if we denote this map by $T$, what does $T$ associates to a tangent vector $v= \frac{d}{dt} \Bigg|_{t=0}[\alpha(t) , \beta(t)] \in T_{[g,X]} M$ ?

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    $\begingroup$ It is more natural to define the isomorphism in the other direction: $\mathfrak{g}/\mathfrak{h} \times \mathfrak{g}/\mathfrak{h} \to T_{[g,X]}(G\times_H \mathfrak{g}/\mathfrak{h})$. It is given by $([Y], [Z]) \mapsto \left.\frac{d}{dt}\right|_{t=0}[ge^{tY},X+tZ]$. $\endgroup$
    – Spenser
    Commented Mar 3, 2022 at 12:46
  • $\begingroup$ Thanks for your comment, but I don't understand why is this map injective ? $\endgroup$
    – Mira
    Commented Mar 4, 2022 at 2:19
  • $\begingroup$ Also could you please explain why it's a well defined map ? $\endgroup$
    – Mira
    Commented Mar 4, 2022 at 3:23

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