The characteristic polynomials of imprimitive groups and affine Coxeter groups

In this paper, we will seek appropriate generators to define the characteristic polynomials of $G(r,1,n)$, and prove that for every finite dimensional representation of $G(r,1,n)$, the characteristic polynomial of $G(r,1,n)$ determines the character of this representation. Furthermore, the same conclusion holds for affine Coxeter groups $\tilde{W}$ and $G(r,p,n)$.