# Show that $(\mu, w\alpha) \leq 0$.

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 $$w(-\alpha) \in \Delta$$?

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 $$\mu \in \overline{C(\Delta)}$$, $$(\beta,\mu)=\sum_{i=1}^n k_i (\alpha_i,\mu) \leq 0.$$