Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 102 / 5 / a / Solution

The weight lattice isFor each root $\alpha$, restrict $V$ to the subalgebraIf $V_{\lambda }\ne 0$, the ${\mathfrak{sl}}_2$ classification shows that the $h_{\alpha }$-eigenvalue $\lambda (h_{\alpha })=⟨\lambda ,{\alpha }^{\vee }⟩$ is an integer. Hence $\lambda \in X$.

The same classification makes every $\alpha$-string symmetric underand preserves weight multiplicity. Since the Weyl group is generated by these simple reflections,