Lemma 7

Tag 0AQI, permanent. Cite it as [Bourbaki, Tag 0AQI].

Lie Groups and Lie Algebras, chapter V, § 3. Groups of displacements generated by reflections

Let $A$ be a set of unit vectors in $T$ . If there exists a real number $\lambda < 1$ such that $(a|a') \leq \lambda$ for $a, a' \in A$ and $a \neq a'$ , then the set $A$ is finite.

For $a, a' \in A$ such that $a \neq a'$ , we have

\| a - a' \| ^2 = 2 - 2(a|a') \geq 2 - 2\lambda.

Now, the unit sphere

S

T

being compact, there exists a finite covering of

S

by sets of diameter

< (2 - 2\lambda)^{1/2}

and each of these sets contains at most one point of

A

, hence the lemma.

Denote by $U(w)$ the automorphism of $T$ associated to the affine map $w \in W$ from $E$ to itself. We have

w(x + t) = w(x) + U(w).t \quad \text{for } t \in T \text{ and } x \in E.

This defines a homomorphism

U

from the group

W

to the orthogonal group of

T

; the kernel of

U

is the set of translations belonging to

W

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Theorem 3, § 3

Cites

Nothing.