Representation Theory – Action of Negative Cartan-Weyl Generators

Question

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.

Answer 1

Let α be a simple root that is not α_k. Associated to this simple root is a subalgebra isomorphic to $\mathfrak{sl}_2$. For this subalgebra, an element of weight ω_k 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 F_α.