Roots of polynomial $\sum_{\sigma \in W} x^{l(\sigma)}$

Question

Let $W$ be Weyl group of a root system $\Phi$ (of finite dimensional simple Lie algebra). For $\sigma\in W$, $l(\sigma)$ be the its length. Consider the following polynomial $$P_\Phi(x) = \sum_{\sigma \in W} x^{l(\sigma)} \in \mathbb{Z}[x]$$

Are all roots of $P_\Phi(x)$ roots of unity?

Some examples:

$\Phi$	$P_\Phi(x)$
$A_2$	$1+2x+2x^2+x^3$
$G_2$	$1+2x+2x^2+2x^3+2x^4+2x^5+x^6$
$C_3$	$1+3x+5x^2+7x^3+8x^4+8x^5+7x^6+5x^7+3x^8+x^9$

One checks easily the answers are yes for them.

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I tried making some specializations in Weyl denominator formula $$\sum_{\sigma \in W} (-1)^{l(\sigma)} e^{\sigma \delta} = e^\delta \prod_{\alpha > 0} (1-e^{-\alpha}), \qquad \delta = \frac{1}{2}\sum_{\alpha>0}\alpha$$ such that LHS becomes $P_\Phi(x)$, but so far not successful.

Answer 1

Yes. Your polynomial $P_\Phi$ is called the Poincaré polynomial, and equals $$P_\Phi(x) = \prod_{i = 1}^n \frac{x^{d_i} - 1}{x - 1},$$ where $d_1, \dotsc, d_n$ are the degrees of the Weyl group $W(\Phi)$. This is, for example, Theorem 3.15 in Humphreys - Reflection groups and Coxeter groups.