# Action on the highest weight vector of a representation of a semisimple linear algebraic group

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.

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It is true. Say $\alpha$ is not perpendicular to $\lambda$. The element of the Weyl group corresponding to $\alpha$ moves $\lambda$, hence $v$. But it has a representative in the subgroup generated by $U_\alpha$, $U_{-\alpha}$. So these cannot both fix $v$. – Wilberd van der Kallen May 1 '11 at 17:04
Thanks, that exactly what I was searching for. – Benjamin Schmidt May 1 '11 at 18:56
You are forgiven for the misplaced apostrophe in the possessive version of my family name (which is commonly spelled in several ways anyway). I didn't actually choose that name, which my illiterate ancestors in England didn't worry about spelling correctly. It seems to have come from an old and now obsolete trade, which is why I took up mathematics. – Jim Humphreys May 5 '11 at 22:27
I'll remember that, if I'm going to cite anything from you in the future :-) – Benjamin Schmidt May 6 '11 at 7:40

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 $\langle \alpha,\lambda\rangle=0$.