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Let $\Phi$ a root system and $W$ the Weyl group relative to basis $\Delta$. Let $\mu \in \overline{C(\Delta)}$, where $C(\Delta)$ is the Weyl chamber fundamental, $w \in W$ and $\alpha \in \Delta$ such that $w\alpha \prec 0$. Show that $(\mu, w\alpha) \leq 0$.

Comments: I need this to understand a demonstration of the Lemma 10.3B of book J.E. Humphreys - Introduction to Lie algebras and representation theory. He states that this fact occurs because $\mu \in \overline{C(\Delta)}$. But I can not see why.

I'm assuming that $(\mu, w\alpha) > 0$. If $\mu \in C(\Delta)$ then $w\alpha \in \Delta \Rightarrow w \alpha \succ 0$ a contradiction. I can not solve the case where $\mu \in \overline{C(\Delta)} \setminus C(\Delta)$.

Another way is as follows: $(\mu, w \alpha) \leq 0 \Leftrightarrow -(\mu, w \alpha) \geq 0 \Leftrightarrow (\mu, w(- \alpha)) \geq 0 \Leftrightarrow w(- \alpha) \in \Delta$. Knowing that $\alpha \in \Delta$ and $w\alpha \prec 0$ I can conclude that $w(-\alpha) \in \Delta$?

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Write $\alpha_1,\dots,\alpha_n$ for the given choice of simple roots. Suppose $\beta$ is a root with $\beta \prec 0$, that is $$\beta=\sum_{i=1}^n k_i \alpha_i,$$ with $k_i \leq 0$. We must show that $(\beta,\mu)$ is non-positive for all $\mu$ in the closure of the fundamental chamber. But this is an immediate consequence of the definition, which we now recall.

The fundamental chamber $C(\Delta)$ is defined by $$C(\Delta)=\{v \in \mathfrak{h} \ | \ (\alpha,v) > 0 \quad \hbox{for all positive roots $\alpha$} \}$$ and its closure is $$\overline{C(\Delta)}=\{v \in \mathfrak{h} \ | \ (\alpha,v) \geq 0 \quad \hbox{for all positive roots $\alpha$} \}.$$ Therefore by definition, for any $\mu \in \overline{C(\Delta)}$, $$(\beta,\mu)=\sum_{i=1}^n k_i (\alpha_i,\mu) \leq 0.$$

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