Let $G$ be a Lie group and $H$ be a Lie subgroup acting on G by right multiplication and acting on $\mathfrak{g}^*$ by the adjoint action.
What is the natural action of H on $\wedge\mathfrak{g}^* ?$
Let $\mathfrak{r}$ be a $H$-invariant subspace of $\mathfrak{g}$ such that $\mathfrak{g}= \mathfrak{h} \oplus \mathfrak{r}$.
Could you please explain these inclusions :
$\wedge^{\max} \mathfrak{r}^* \hookrightarrow [\wedge \mathfrak{g}^*]_{H\text{-basic}}$,
$[\wedge \mathfrak{g}^*]_{H\text{-basic}} \hookrightarrow A^*(G/H) $.
I greatly appreciate your help.