Definition 2

The integers $m_1, m_2, \ldots, m_l$ are called the exponents of W.

Let C be a chamber determined by $\mathfrak{H}, H_1, \ldots, H_l$ its walls, and put $s_i = s_{H_i}$. Denote by $e_i$ the unit vector orthogonal to $H_i$ and on the same side of $H_i$ as C. By Prop. 2 of the Appendix of Chap. IV, we can assume that the $H_i$ are numbered so that $e_1, e_2, \ldots, e_r$ are pairwise orthogonal and $e_{r+1}, e_{r+2}, \ldots, e_l$ are pairwise orthogonal. Then $s' = s_1 s_2 \ldots s_r$ is the orthogonal symmetry with respect to the subspace

$s'' = s_{r+1} s_{r+2} \ldots s_l$ is the orthogonal symmetry with respect to the subspace

and $c = s' s''$ is a Coxeter transformation. Since $(e_1, \ldots, e_l)$ is a basis of V, V is the direct sum of $V'$ and $V''$.

We deduce first that 1 is not an eigenvalue of c. For if $x \in V$ is such that $c(x) = x$, then $s'(x) = s''(x)$, so $x - s'(x) = x - s''(x)$ is orthogonal to $V'$ and $V''$, and hence is zero. Thus, $x = s'(x) = s''(x) \in V' \cap V'' = \{0\}$.

Consequently,

(1)

The characteristic polynomial of c has real coefficients. Thus, for all $j$, the power of $T - \exp \frac{2i\pi m_j}{h}$ in P(T) is equal to that of $T - \exp \frac{2i\pi (h - m_j)}{h}$. Hence

(2)

Adding the equalities (2) we obtain

(3)