Suppose that $G$ is a complex semisimple Lie group, $P$ a parabolic subgroup of $G$. What are all of the $P$-invariant subspaces of $\mathfrak{g}/\mathfrak{p}$? In various low dimensional examples, I can calculate them all out, but there should be some easy way to describe each of them in terms of the Dynkin diagram of $G/P$.

Suppose that $G$ is a complex semisimple Lie group, $P$ a parabolic subgroup of $G$. What are all of the $P$-invariant subspaces of $\mathfrak{g}/\mathfrak{p}$? In various low dimensional examples, I can calculate them all out, but there should be some easy way to describe each of them in terms of the Dynkin diagram of $G/P$. We can see how complicated the associated graded gets in two examples: each irreducible $P$-module of the associated graded is a connected component in the picture.