Let $G$ be a semisimple linear algebraic group, $V$ a $G$-representation and $v \in V$ a vector of highest weight $\lambda$. Is it true, that for any positive root $\alpha \in R^+$ the one dimensional unipotent subgroup $U_{-\alpha}$ acts trivially on $v$ if and only if $\langle \alpha^{\vee}, \lambda \rangle = 0$?

I'm asking, because I want to apply this to the following situation. For any simple root $\alpha$, I denote the corresponding fundamental weight by $\omega$. Let $V=V(\omega)$ be the simple representation of highest weight $\omega$ with highest weight vector $v$. I'm trying to prove, that the stabilizor of the element $[v] \in \mathbb{P}(V)$ is the maximal parabolic that doesn't contain $U_{-\alpha}$.

The only reference I have for representations of semisimple groups is the small chapter in Humphrey's book Linear algebraic groups. I would also be grateful, if someone could give me a bigger reference on that subject.