Let $G$ be a Lie group and $H$ be a subgroup such that $G/H$ is also a Lie group. How do I describe left-invariant differential forms on $G/H$? Thinking in terms of Lie algebras, it should be the dual of $\mathfrak{g}/\mathfrak{h}$, where $\mathfrak{g}$ is the Lie algebra of $G$ and $\mathfrak{h}$ is the Lie algebra of $H$.
But I was looking for some characterisation such as "a differential form on $G/H$ is a differential form on $G$ invariant under the action of $H$". I think it is true, more generally, that if $M$ is a manifold and $H$ is a group with a proper free action on $M$, then differential forms on $M/H$ correspond to $H$-invariant differential forms on $M$.
I'm a bit confused with the fact that left-invariant differential forms on $G$ (i.e., elements of $\mathfrak{g}$) are by definition invariant under the whole group (in particular, they are also $H$-invariant). Are there two different notions of invariance?