Action of a Lie subgroup

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 $wedgemathfrak{g}^* ?$

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 :

$wedge^{max} mathfrak{r}^* hookrightarrow [wedge mathfrak{g}^*]_{Htext{-basic}}$,

$[wedge mathfrak{g}^*]_{Htext{-basic}} hookrightarrow A^*(G/H) $.

I greatly appreciate your help.