Definition 2

An element $x \in A[P]$ is said to be anti-invariant under $W$ if

for all $w \in W$.

The anti-invariant elements of $A[P]$ form an $A$-submodule of $A[P]$. For any $x \in A[P]$, put

(1)

For $x \in A[P]$ and $w \in W$, we have

and $J(x)$ is anti-invariant. On the other hand, let $q = \mathrm{Card}(W)$. For any anti-invariant element $x$ of $A[P]$, we have $J(x) = q.x$. It follows that, if $q$ is invertible in $A$, the map $q^{-1}J$ is a projection from $A[P]$ onto the submodule of anti-invariant elements.

Let $\overline{\omega}_1, \ldots, \overline{\omega}_l$ be the fundamental weights corresponding to the chamber $C$. The elements of $P \cap \overline{C}$ (resp. $P \cap C$) are the weights of the form $n_1 \overline{\omega}_1 + \cdots + n_l \overline{\omega}_l$ with $n_i \geq 0$ (resp. $n_i > 0$) for $1 \leq i \leq l$ (§ 1, no. 10). On the other hand,

is half the sum of the positive roots (loc. cit.) so the elements of $P \cap C$ are the weights of the form $\rho + p$ with $p \in P \cap \overline{C}$. Finally, if $p \in P \cap C$, then $w(p) < p$ for all $w \neq 1$ (§ 1, no. 6, Cor. to Prop. 18) and $e^p$ is thus the unique maximal term of $J(e^p)$.