Proposition 1

If 2 is not a zero divisor in $A$, the elements $J(e^p)$ for $p \in P \cap C$ form a basis of the module of anti-invariant elements of $A[P]$.

The weights $w(p)$ for $w \in W$ and $p \in P \cap C$ are pairwise distinct. It follows that the $J(e^p)$ for $p \in P \cap C$ are linearly independent.

On the other hand, let $x = \sum_p x_p e^p$ be an anti-invariant element of $A[P]$. If $p_0$ belongs to a wall, it is invariant under a reflection $s \in W$ and

It follows that $2x_{p_0} = 0$, so $x_{p_0} = 0$. Since every element that does not belong to any wall can be written uniquely in the form $w(p)$ with $w \in W$ and $p \in P \cap C$, we thus have Since $w(x) = \sum_p x_p e^{w(p)} = \varepsilon(w) \sum_p x_p e^p$, $x_{w(p)} = \varepsilon(w)x_p$ and we deduce from (2) that which completes the proof.

Consider now the element $d$ of the algebra $A[\frac{1}{2}P]$ defined by

Since $\rho \in P$, $d \in A[P]$.