We know that $W=N/T$ ($N = \{n \in G \mid nTn^{-1} = T \}$) acts on $T$ by $w(t)=wtw^{-1}$ (since $T$ is commutative, this action is well-defined). In the case of $GL_n$, by direct computation we know that $w_0(\alpha_i^{\vee}(c))=\alpha_{i^*}^{\vee}(c^{-1})$, where $w_0$ is the longest element in $W$, $\alpha_i^{\vee}$ is a coroot, $i^*$ is $\tau$ and $\tau$ is the automorphism of the Dynkin diagram. My question is how to prove $w_0(\alpha_i^{\vee}(c))=\alpha_{i^*}^{\vee}(c^{-1})$ in other types other than type A (in type A, $GL_n$ case we can use matrices to verify). It is said that this is well-know. But I am not able to find a reference. Are there some references about this fact? Thank you very much.
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When you say $i^\ast = \tau$ is the automorphism of the Dykin diagram what do you mean? Is there a specific one you have in mind or just there there exists one? – Jim Dec 19 '13 at 19:41
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@Jim, thank you very much. It is a specific one. For example, in the case of A2, 1^* = 2 and 2^* = 1. – LJR Dec 20 '13 at 08:58