Proposition 2

(i) The element d defined by (3) is an anti-invariant element of A[P]; its unique maximal term (no. 2, Def. 1) is $e^{\rho}$ and $d = J(e^{\rho})$. (ii) For any $p \in P$, the element $J(e^p)$ is divisible uniquely by $d$ and the quotient $J(e^p)/d$ is an element of $A[P]$ invariant under $W$. (iii) If 2 is not a zero divisor in $A$, multiplication by $d$ is a bijection from the set of elements of $A[P]$ invariant under $W$ to the set of anti-invariant elements of $A[P]$.

We know that, for $1 \leq i \leq l$, the reflection $s_i = s_{\alpha_i}$ leaves stable the set of positive roots other than $\alpha_i$ and that $s_i(\alpha_i) = -\alpha_i$ (§ 1, no. 6, Cor. 1 of Prop. 17). Hence,

Since the $s_i$ generate $W$, this proves the first assertion in (i). The second assertion in (i) follows immediately from (3) and Lemma 2, noting that 1 is the unique maximal term of $1 - e^{-\alpha}$ for $\alpha \in R, \alpha > 0$.

Assume now that $A = \mathbf{Z}$. By Prop. 1,

On the other hand, it is clear that

If $p \in P \cap C$ with $p \neq \rho$, then $p > \rho$ and the coefficient of $e^p$ in $d$ is zero by (5). Thus, $c_p = 0$. Moreover, comparison of the coefficients of $e^\rho$ in (4) and (5) shows that $c_\rho = 1$ and hence that $d = J(e^\rho)$.

We continue to assume that $A = \mathbf{Z}$. Let $p \in P, \alpha \in \mathbf{R}$ and $M$ be a system of representatives of the right cosets of $W$ with respect to the subgroup $\{1, s_\alpha\}$. Then,

Now $s_\alpha w(p) = w(p) - \langle \alpha^*, w(p) \rangle \alpha = w(p) + n_w \alpha$, with $n_w \in \mathbf{Z}$. Thus,

If $n_w \geq 0$, it is clear that $1 - e^{n_w \alpha}$ is divisible by $1 - e^\alpha$ and this is also true when $n_w < 0$ since $1 - e^{n_w \alpha} = -e^{n_w \alpha}(1 - e^{-n_w \alpha})$. Hence, $J(e^p)$ is divisible by $1 - e^\alpha$ in $\mathbf{Z}[P]$.

By Lemma 1, $\mathbf{Z}[P]$ is factorial and the elements $1 - e^\alpha$ for $\alpha \in \mathbf{R}$ and $\alpha > 0$ are mutually prime. It follows that $J(e^p)$ is divisible in $\mathbf{Z}[P]$ by the product $\prod_{\alpha > 0} (1 - e^\alpha)$, and hence also by $d = e^{-\rho} \prod_{\alpha > 0} (1 - e^\alpha)$.

Returning to the general case, by extension of scalars from $\mathbf{Z}$ to $A$, we deduce from the above that $d = J(e^\rho)$ and that every element $J(e^p)$ is divisible by $d$. Since $e^\rho$ is the unique maximal term of $d$, the Remark of no. 2 shows that there exists a unique element $y \in A[P]$ such that $J(e^p) = dy$ and it follows immediately that $y$ is invariant under $W$, and hence that $d$ and $J(e^p)$ are anti-invariant. This proves (i) and (ii).

Finally, if 2 is not a zero divisor in $A$, the Remark of no. 2 and Prop. 1 imply (iii).