Minuscule weights have exactly the Weyl orbit as their weights

First suppose that $\omega$ is minuscule and nonzero; by [F1] it is a fundamental weight ${\omega }_i$. We argue by induction on the rank of the irreducible root system. Let $\lambda$ be dominant integral and write $\beta :={\omega }_i-\lambda ={\sum }_j{m}_j{\alpha }_j\in Q_{+}$. If some $k\mathrm{\ne }i$ has $m_k=0$, delete the vertex $k$ from the Dynkin diagram. By [F6], the root subsystem generated by the remaining simple roots is the orthogonal direct sum of the subsystems ${\mathrm{\Phi }}_C$ for the connected components $C$ of the deleted diagram; their spans are mutually orthogonal, and $\beta$ is the sum of its component projections ${\beta }_C$. In each component not containing $i$, the projection of ${\omega }_i$ is zero, so ${\beta }_C=-{\lambda }_C$, where ${\lambda }_C$ is dominant for that component. Writing ${\beta }_C={\sum }_{j\in C}{m}_j{\alpha }_j$, we have $({\beta }_C,{\beta }_C)=-{\sum }_{j\in C}{m}_j({\lambda }_C,{\alpha }_j)\le 0$ because $({\lambda }_C,{\alpha }_j)=\frac{1}{2}({\alpha }_j,{\alpha }_j)⟨{\lambda }_C,{\alpha }_j^{\vee }⟩\ge 0$. Positive definiteness gives ${\beta }_C={\lambda }_C=0$. The component $C_i$ containing $i$ has smaller rank; the projection of ${\omega }_i$ is its fundamental weight, still minuscule, and ${\lambda }_{C_i}$ is dominant integral with ${\omega }_i{\mathrm{\mid }}_{C_i}-{\lambda }_{C_i}={\beta }_{C_i}\in Q_{+}(C_i)$. The induction hypothesis applies to the irreducible lower-rank system ${\mathrm{\Phi }}_{C_i}$ and gives ${\beta }_{C_i}=0$, hence $\beta =0$. Thus, if $\beta \mathrm{\ne }0$, then $m_j>0$ for every $j\mathrm{\ne }i$.

A root-lattice element with all pairings bounded by $1$ vanishes: if $\xi \in Q$ satisfies $\mathrm{\mid }⟨\xi ,{\beta }^{\vee }⟩\mathrm{\mid }\le 1$ for every coroot ${\beta }^{\vee }$, then $\xi =0$. Suppose not, and choose a counterexample $\xi ={\sum }_k{m}_k{\alpha }_k$ with ${\sum }_k\mathrm{\mid }{m}_k\mathrm{\mid }$ minimal. Then $(\xi ,\xi )={\sum }_k{m}_k(\xi ,{\alpha }_k)>0$ by positive definiteness, so some $k$ has $m_k\mathrm{\ne }0$ and $(\xi ,{\alpha }_k)$ of the same sign as $m_k$; replacing $\xi$ by $-\xi$ if necessary, we may assume $m_k>0$ and $(\xi ,{\alpha }_k)>0$, so $⟨\xi ,{\alpha }_k^{\vee }⟩=2(\xi ,{\alpha }_k)\mathrm{/}({\alpha }_k,{\alpha }_k)=1$ by the bound. Then $s_k\xi =\xi -{\alpha }_k$ is again a counterexample, since $\mathrm{\mid }⟨s_k\xi ,{\beta }^{\vee }⟩\mathrm{\mid }=\mathrm{\mid }⟨\xi ,s_k{\beta }^{\vee }⟩\mathrm{\mid }\le 1$ for all coroots (the set of coroots is $W$-stable), and it has coordinate sum ${\sum }_j\mathrm{\mid }{m}_j\mathrm{\mid }-1$, contradicting minimality. Hence $\xi =0$.

The weight set of a finite-dimensional module is $W$-stable. Let $M$ be such a module, let $\mu$ be a weight, and let $\alpha ={\alpha }_k$ be simple with $t=⟨\mu ,{\alpha }^{\vee }⟩$. The subalgebra ${\mathfrak{s}\mathfrak{l}}_2(\alpha )$ acts on $M$. Decompose it into irreducibles and write a nonzero weight vector $v$ as the sum of its components in their weight-$t$ spaces. In each irreducible summand where that component is nonzero, the highest weight is some $m\ge \mathrm{\mid }t\mathrm{\mid }$ with $m\equiv t(\mathrm{m}\mathrm{o}\mathrm{d}2)$. If $t\ge 0$, lowering that component by $t$ steps is nonzero and has weight $-t$; if $t<0$, raising it by $-t$ steps is nonzero and has weight $-t$. Their direct sum is nonzero and has weight $\mu -t\alpha =s_k\mu$. Thus each simple reflection preserves the set of weights, and these reflections generate $W$.

(2)$\Rightarrow$(1): suppose (2) holds and $\omega$ is not minuscule. Then by [F1] there is a positive root $\alpha$ with $⟨\omega ,{\alpha }^{\vee }⟩\ge 2$. Let $v_{\omega }\mathrm{\ne }0$ be a highest weight vector; the root vector $x_{-\alpha }$ of the ${\mathfrak{s}\mathfrak{l}}_2(\alpha )$-triple [F5] satisfies $x_{-\alpha }{v}_{\omega }\mathrm{\ne }0$, because $v_{\omega }$ is annihilated by all positive root vectors and spans the highest weight space of the ${\mathfrak{s}\mathfrak{l}}_2(\alpha )$-module it generates, of highest weight $⟨\omega ,{\alpha }^{\vee }⟩\ge 1$; this vector has weight $\omega -\alpha$ (Root vectors shift weights). By (2) there is $w\in W$ with $\omega -\alpha =w\omega$, and the $W$-invariance of the form [F2] gives $(\omega -\alpha ,\omega -\alpha )=(\omega ,\omega )$, while $(\omega -\alpha ,\omega -\alpha )=(\omega ,\omega )-2(\omega ,\alpha )+(\alpha ,\alpha )<(\omega ,\omega )$ because $2(\omega ,\alpha )=⟨\omega ,{\alpha }^{\vee }⟩(\alpha ,\alpha )\ge 2(\alpha ,\alpha )>(\alpha ,\alpha )$. This contradiction proves (2)$\Rightarrow$(1).

Suppose $⟨{\omega }_i-\lambda ,{\alpha }_i^{\vee }⟩\le 0$. For every $j\mathrm{\ne }i$, $⟨{\omega }_i-\lambda ,{\alpha }_j^{\vee }⟩=-⟨\lambda ,{\alpha }_j^{\vee }⟩\le 0$, since ${\omega }_i$ is the $i$th fundamental weight and $\lambda$ is dominant. Together with the assumed nonpositivity at $i$, all simple-coroot pairings of $\beta ={\omega }_i-\lambda ={\sum }_k{m}_k{\alpha }_k$ are nonpositive. Therefore $(\beta ,\beta )={\sum }_k{m}_k(\beta ,{\alpha }_k)={\sum }_k{m}_k\frac{({\alpha }_k,{\alpha }_k)}{2}⟨\beta ,{\alpha }_k^{\vee }⟩\le 0$, because each $m_k\ge 0$. Positive definiteness gives $\beta =0$ and $\lambda ={\omega }_i$.

It remains to exclude the case $⟨{\omega }_i-\lambda ,{\alpha }_i^{\vee }⟩>0$, which is $1$ because ${\omega }_i$ pairs by $1$ and $\lambda$ is dominant integral. Write $\beta ={\omega }_i-\lambda ={\sum }_k{m}_k{\alpha }_k$. Then $m_k>0$ for $k\mathrm{\ne }i$ by step 1.1, and the Cartan-integer formula gives $1=⟨\beta ,{\alpha }_i^{\vee }⟩=2m_i+{\sum }_{j\mathrm{\ne }i}{m}_j⟨{\alpha }_j,{\alpha }_i^{\vee }⟩$. The off-diagonal Cartan integers are nonpositive, so $2m_i\ge 1$ and the integer $m_i$ is positive; hence every $m_j\ge 1$. Let $\theta$ be the highest root and write ${\theta }^{\vee }={\sum }_j{c}_j{\alpha }_j^{\vee }$ with all $c_j>0$ as in [F1]. By Existence and uniqueness of the highest root, $(\theta ,{\alpha }_j)\ge 0$ for every simple root. Since ${\theta }^{\vee }$ is a positive scalar multiple of $\theta$, this gives $⟨{\alpha }_j,{\theta }^{\vee }⟩=2({\alpha }_j,\theta )\mathrm{/}(\theta ,\theta )\ge 0$; at least one is positive because the simple roots span and $\theta \mathrm{\ne }0$. Therefore $\beta ({\theta }^{\vee })={\sum }_j{m}_j⟨{\alpha }_j,{\theta }^{\vee }⟩>0$. By [F1], $⟨{\omega }_i,{\theta }^{\vee }⟩=c_i=1$, so $\lambda ({\theta }^{\vee })=1-\beta ({\theta }^{\vee })<1$. This is a nonnegative integer because $\lambda$ is dominant integral and every $c_j>0$; hence it is zero, all simple-coroot pairings of $\lambda$ vanish, and $\lambda =0$. Thus $\beta ={\omega }_i$; since $\beta$ was in $Q_{+}$, this proves ${\omega }_i\in Q$ before applying the root-lattice lemma.

(3)$\Rightarrow$(2): let $\mu$ be a weight of $L(\omega )$. By [F4] choose $w\in W$ with $\lambda :=w\mu$ dominant; by step 1.3 and induction on a decomposition of $w$ into simple reflections, $\lambda$ is again a weight of $L(\omega )$, hence $\omega -\lambda \in Q_{+}$ by [F3]. Assumption (3) gives $\lambda =\omega$, so $\mu =w^{-1}\omega \in W\omega$.

Conclusion of (1)$\Rightarrow$(3): if the case of step 2.2 occurs, it gives $\lambda =0$ and $\beta ={\omega }_i\in Q$. The nonzero minuscule weight ${\omega }_i$ has all coroot pairings bounded in absolute value by $1$, so step 1.2 now applies and forces ${\omega }_i=0$, a contradiction. Together with step 2.1, this proves $\beta =0$ and $\lambda ={\omega }_i=\omega$. If $\omega =0$ and $-\lambda ={\sum }_k{m}_k{\alpha }_k\in Q_{+}$ with $m_k\ge 0$, then $(\lambda ,\lambda )=-{\sum }_k{m}_k(\lambda ,{\alpha }_k)\le 0$ because $\lambda$ is dominant; positive definiteness gives $\lambda =0$. This proves (1)$\Rightarrow$(3).

Consequences. Assume $\omega$ is minuscule. By (2) every weight of $L(\omega )$ lies in $W\omega$, and by [F3] every element of $W\omega$ occurs with multiplicity exactly one; hence every weight space is one-dimensional (in particular at most one-dimensional), there are exactly $\mathrm{\mid }W\omega \mathrm{\mid }$ distinct weights, and $\mathrm{ch}L(\omega )={\sum }_{\gamma \in W\omega }{e}^{\gamma }$.