Let $\frak{g}$ be a complex simple Lie algebra of rank $l$. For $\frak{h}$ a choice of Cartan subalgebra, let $\alpha\_1, \cdots, \alpha\_r$ be the corresponding choice of simple roots, $X\_{\alpha\_i}, H\_{\alpha\_i}, X\_{-\alpha\_i}$ the Cartan--Weyl basis, and $\pi\_1, \cdots, \pi\_l$ the fundamental weights. For the irreducible $\frak{g}$-module $V(\pi\_k)$ let $v \in V(\pi\_k)$ be a highest weight vector, i.e.
$$
X\_{\alpha\_i}v = 0 ~~ \forall i=1, \dots, r.
$$
How will the elements $X\_{-\alpha\_i}$ act on $v$? Initial experiments suggest that
$$
X\_{-\alpha\_i}v = 0, ~~ \forall i \neq k.
$$
However I cannot seem to see a general proof. Any help is very appreciated.



Action of the negative Cartan-Weyl generators on a highest weight element
Let α be a simple root that is not αk. Associated to this simple root is a subalgebra isomorphic to $\mathfrak{sl}\_2$. For this subalgebra, an element of weight ωk has weight zero. In the fundamental representation you are considering, such an element is highest weight by assumption and therefore by the classificaition of representations of $\mathfrak{sl}\_2$ is annihilated by Fα.


