Let $d_1, \dotsc d_n$ be the degrees (of fundamental invariants) of the Weyl group $W$ of a simple Lie group, (in the reflection representation; see table given on [the Wikipedia page](https://en.wikipedia.org/wiki/Coxeter_element) for their explicit values). 

If a positive integer $d$ is a degree of $W$, then so is $h+2-d$, where $h$ is the Coxeter number associated to $W$. This suggests considering the *real* polynomial 
$$P_W(x) := \prod_{j=1}^n \left(x - \omega^{d_j}\right),$$

where $\omega = \exp(\frac{2\pi i}{h+2})$. 

For instance, in type $G_2$, the degrees are $2$ and $6$, $h+2 = 8$ and so 
$$P_{G_2}(x) = x^2+1.$$

**Precise question:** Is there a reference in the literature for the expanded form of these polynomials for all types $ABCDEFG$?

**Vague question:** Has anyone studied such polynomials? In particular, is there a way of seeing them arise naturally (e.g. as characteristic polynomials, or other…)?

**Update:**
One setting in which $P_W(x)$ makes an appearance is as follows. Let $\Phi$ denote the set of roots associated to $W$, and let ${\rm ht}: \Phi \longrightarrow \mathbb{Z}$ denote the height function, where by convention, ht$(-\beta) = - {\rm ht}(\beta)$ for a positive root $\beta$. Then, one has the rather surprising
$$
(x^{h+2}-1)^n = (x-1)^n \, P_W(x) \, R_W(x),
$$  
where $R_W(x) = \prod_{\beta \in \Phi}(x - \omega^{{\rm ht}(\beta)})$. I don't know the "meaning" of this fact, though.. 

For instance, in type $A_3$, there are 3 roots of height $1$, 2 roots of height $2$ and 1 root of height $3$. Thus with $\omega = \exp(\frac{2\pi i}{6})$, we get
\begin{align*}
R_{A_3}(x) &= (x-\omega)^3 \, (x-\omega^2)^2 \, (x-\omega^3) \, (x-\omega^{-1})^3 \, (x-\omega^{-2})^2 \, (x-\omega^{-3}) \\
&= (x-\omega)^3 \, (x-\omega^2)^2 \, (x-\omega^3)^2 \, (x-\omega^4)^2 \, (x-\omega^5)^3,
\end{align*}
and 
$$P_{A_3}(x) = (x-\omega^2) (x-\omega^3) (x-\omega^4),
$$
from which it is clear that $(x^6-1)^3 = (x-1)^3 P_{A_3}(x) R_{A_3}(x).$