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.



Action on the highest weight vector of a representation of a semisimple linear algebraic group
For any irreducible representation $V$ with highest weight vector $v$ and highest weight $\lambda$, the stabilizer of $[v]\in \mathbb{P}V$ is the parabolic subgroup corresponding to those simple roots that are orthogonal to $\lambda$. By this I mean the parabolic generated by $\mathfrak{h}$, all positive root spaces, and the negative root spaces $\mathfrak{g}\_{-\alpha}$ with $\alpha$ simple and $\langle \alpha,\lambda\rangle=0$.


For the representation theory of semisimple algebraic groups I strongly recommend the book "Representation Theory" by Fulton and Harris; in particular, this answer was taken from §23.3, Homogeneous spaces, p. 382–395 (see Claim 23.52 and the following discussion).


