Let $G$ be a compact Lie group and let $H$ be a closed subgroup of $G$, with Lie algebras $\mathfrak{g}$ and $\mathfrak{h}$.

We denote $G\times_H \mathfrak{g} / \mathfrak{h}$: the set of orbits $(G \times \mathfrak{g} / \mathfrak{h})/H $ of the right action of $H$ on $ G \times \mathfrak{g} / \mathfrak{h}$ ($H$ acts on $G$ by multiplication and acts on $\mathfrak{g} / \mathfrak{h}$ by the adjoint action).

We have this identification: $T(G/H) \simeq G\times_H \mathfrak{g} / \mathfrak{h}$.

My question is why the space of differential forms of $G/H$, $A^\bullet(G/H)$, satisfies $$ A^\bullet(G/H) \simeq \Gamma (G/H, G\times_H {\bigwedge}^\bullet{(\mathfrak{g} / \mathfrak{h})}^*) \simeq {(C^\infty (G) \otimes {\bigwedge}^\bullet {(\mathfrak{g} / \mathfrak{h})}^*)}^H. $$

`G \times H`

, rather than Unicode, like $G × H$`G × H`

. I have edited accordingly. $\endgroup$