Let $\mathfrak{g}$ a semisimple Lie algebra with a Cartan subalgebra $\mathfrak{h}$. Let $\phi$ a root system with base $\Delta$ And $V$ an irreducible representation of $\mathfrak{g}$. Let $\Gamma \subset \mathfrak{h}^*$ finite set, such that $V= \bigoplus_{\gamma \in \Gamma} V_{\gamma}$ , where $V_{\gamma} = \left\{ v \in V | hv = \gamma(h)v \right\}$ (a weight space). I want to prove the following:
- $\Gamma \subset \gamma + \sum_{i=1}^{r} \mathbb{Z}\alpha_i$, with $\alpha_i \in \Delta$
- if $\lambda \in \Gamma$ is a maximal weight for $V$,i.e. such that $\Gamma \in \lambda - \sum_{i=1}^{r} \mathbb{Z}_{\geq 0}\alpha_i$, prove that $\langle \lambda, \alpha_i \rangle$ is a positive integer for every $i= 1...r$
So far, I have only managed to notice that to prove 2) it may be enough to show that $(\lambda, \alpha_i) \geq 0$ for every $i$. By contradiction, if there exists $i$ such that $(\lambda, \alpha_i) < 0$ I could infer that $\lambda + \alpha_i \in \Gamma$, if $\alpha_i \in \Gamma$. Therefore I may have found a contradiction since $\lambda$ is maximal. I don’t know if that makes sense, but if it does, I still don’t know how I could prove that at $\alpha_i \in \Gamma$. Any hints?
Note that if $v$ is such that $hv = \gamma(h)v$, and $e_\alpha$ is a basis of the root space to $\alpha$ in $\mathfrak g$, then
$$h(e_\alpha v) = [h, e_\alpha]v + e_\alpha(hv) = (\alpha(h)\cdot e_\alpha) v + e_\alpha(\gamma(h)v) = \left(\alpha(h) + \gamma(h)\right) (e_\alpha v)$$
i.e. (and this is how I "knew" this before I did the computation):
$$v \in V_\gamma \implies e_\alpha v \in V_{\alpha+\gamma} \qquad (*)$$
To conclude for no.1 from there, proceed as follows: W.l.o.g. there exists $\gamma \in \Gamma$ such that $V_\gamma \neq 0$. Let $0 \neq v \in V_\gamma$, and let $W :=$ the subrepresentation of $V$ generated by $v$, i.e. all linear combinations of all $xv$ for all $x \in \mathfrak g$. Use $(*)$ and the root space decomposition of $\mathfrak g$ to show that $$W \subseteq \bigoplus_{\beta \in \sum \mathbb Z \alpha_i} V_{\gamma + \beta}$$
After that, remember we have not used irreducibility yet; do it now.
For no.2, you should first familiarize yourself with how $\langle \cdot, \cdot\rangle$ is defined.