X = {λ \in t^{*} : ⟨ λ, α^{\lor} ⟩ \in Z for every α \in Φ} .

For each root

α

, restrict

V

m_{α} = span {e_{α}, h_{α}, f_{α}} ≅ sl_{2} .

V_{λ} \neq = 0

, the

sl_{2}

classification shows that the

h_{α}

λ (h_{α}) = ⟨ λ, α^{\lor} ⟩

is an integer. Hence

λ \in X

The same classification makes every

α

-string symmetric under

λ ⟼ λ - ⟨ λ, α^{\lor} ⟩ α = s_{α} λ

and preserves weight multiplicity. Since the Weyl group is generated by these simple reflections,

V_{w λ} \neq = 0 and dim V_{w λ} = dim V_{λ} (w \in W) .

Solved by gpt-5.6-sol high.

 Ancestors (10)

 Discussion (0)

There are no discussions about this article yet.

There are currently no matching articles.