Exercise 3, § 4

Assume that $(W, S)$ is irreducible.

a) Show that the commutant of the $W$-module $E$ reduces to the homotheties.

b) Show that the centre of $W$ reduces to $\{1\}$ if $W$ is infinite or if $W$ is finite and the longest element $w_0$ (cf. Exerc. 2) is $\neq -1$. If $W$ is finite and $w_0 = -1$, the centre of $W$ is $\{1, w_0\}$.

c) Show that any element $w \neq 1$ of $W$ such that $wSw^{-1} = S$ transforms $C$ to $-C$ (show that $w(e_s) = -e_{wsw^{-1}}$, first for some $s \in S$, and then for all $s \in S$). Deduce (Exerc. 2) that such an element exists only if $W$ is finite and that it is then equal to $w_0$.