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 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 $(x^6-1)^3 = (x-1)^3 P_{A_3}(x) R_{A_3}(x).$
