Let $G$ be a connected, simply-connected complex semisimple linear algebraic group with Lie algebra $\frak{g}$. Fix a maximal torus $T\subseteq G$ and let $\Delta\subseteq Hom(T,\mathbb{C}^*)$ be the resulting collection of roots. Choose positive and negative roots, $\Delta_{+}$ and $\Delta_{-}$, respectively. Let $\Pi\subseteq\Delta_{+}$ be the set of simple roots. Let $V(\lambda)$ be the irrep of $G$ with highest (dominant) weight $\lambda$. We know that $$V(\lambda)=U(\frak{n}_-) v_{\lambda},$$ where $v_{\lambda}\in V(\lambda)$ is a high-weight vector.

Is there some characterization of those $\alpha\in\Delta_{-}$ for which $\{0\}=\frak{g}_{\alpha}\cdot v_{\lambda}$? (ie. Which negative root spaces annihilate $v_{\lambda}$?) I think we might generally obtain more than just those $\alpha\in\Delta_{-}$ for which $\lambda+\alpha$ is not a weight of $V(\lambda)$. If this is documented in the literature, I would appreciate any references. If not, I would appreciate any thoughts.