Let $\mathfrak{g}$ be a complex simple Lie algebra, and $W$ its Weyl group. We make a choice of simple roots, and note $\Phi_+$ the corresponding set of positive roots, while $\Phi$ is the set of all roots. Finally, I denote $z^{\lambda}$ the character associated to a weight $\lambda$.

I claim that $$\sum\limits_{w \in W} \prod\limits_{\alpha \in \Phi_+} (1-z^{w(\alpha)}) = \prod\limits_{\alpha \in \Phi}(1-z^{\alpha}) \, . $$

My questions are:

- Where can I find this formula (I would particularly appreciate a textbook) ? Does it have a specific name ?
- If the first question can't be answered, how would you prove the formula ?