Let \((W,S)\) be a Coxeter system, \(r = (r_s)_{s\in S}\) a system of indeterminates such that \(r_s = r_{s'}\) if and only if \(s\) and \(s'\) are \(W\)-conjugate and different \(r_s\)’s are algebraically independent, \(R = \mathbb{Z}[r]\), and \(H(R)\) a free \(R\)-module with a basis \(\{T_W\}_{w \in W}\). Let \(F\) be a field, \(O : R \to F\) a ring homomorphism, and assume \(W\) finite. The author shows that \(H(R) \otimes F\) is semisimple if and only if the specialization by \(O\) of every generic degree divided by the Poincaré polynomial is well-defined.