Proposition 5

Let $J_1, \ldots, J_s$ be the sets of vertices of the connected components of the Coxeter graph of W. For $1 \leq p \leq s$, put

(i) The group W is the direct product of the groups $W_p$ ($1 \leq p \leq s$). (ii) The space T is the orthogonal direct sum of the subspaces $T_0, T_1, \ldots, T_s$, which are all stable under U(W). (iii) For all q such that $1 \leq q \leq s$, the subspace $T'_q$ of T consists of the vectors invariant under U(Wq); it is the direct sum of the $T_p$ for $1 \leq p \leq s$ and $p \neq q$.*

(iv) Let C be a chamber. The subspace T_p (for $1 \leq p \leq s$) is generated by the vectors $e_i(C)$ for $i \in J_p$ (in the notation of no. 4).

(v) The representations of W in the subspaces T_p ($1 \leq p \leq s$) are absolutely irreducible, non-trivial, and pairwise inequivalent.

Assertion (i) follows from Chap. IV, §1, no. 9. On the other hand, we have already seen that the subspaces T_p are invariant under U(W), and so is T_0. Let C be a chamber; since W_p is generated by the reflections $s_i(C)$ for $i \in J_p$, it is clear that $T'_p$ is the subspace orthogonal to the $e_i(C)$ for $i \in J_p$, hence (iv). Moreover, if $i \in J_p, j \in J_q$ with $p \neq q$, then $m_{ij} = 2$ since $\{i, j\}$ is not an edge of the Coxeter graph of W, so $(e_i|e_j) = 0$. Assertion (ii) is now immediate. And assertion (iii) follows, since $T'_q$ is the orthogonal complement of $T_q$.

Finally, let V be a subspace of T_p invariant under U(W_p). For all $i \in J_p$, either $e_i \in V$ or $e_i$ is orthogonal to V (§ 2, no. 2, Prop. 3). Let A (resp. B) be the set of $i \in J_p$ such that $e_i \in V$ (resp. $e_i$ is orthogonal to V). Clearly $(e_i|e_j) = 0$ for $i \in A$ and $j \in B$, and since $J_p$ is connected, it follows that either $A = \varnothing$ and $V = \{0\}$, or $A = J_p$ and $V = T_p$. Consequently, the representation of W_p on T_p is irreducible, hence absolutely irreducible (§ 2, no. 1, Prop. 1). It is non-trivial by the very definition of T_p. Finally, the last assertion in (v) follows immediately from (iii).

If the subspace T_0 of vectors in T invariant under U(W) reduces to $\{0\}$, then W is said to be essential; if the representation U of W on T is irreducible, then W is said to be irreducible.