A question on irreducible affine Coxeter groups

Question

I have a question about affine Coxeter groups when reading Humphreys's book:

Let $(W,S)$ be an irreducible affine Coxeter group, $M=(m_{ij})$ be its Coxeter matrix, and $\{\alpha_s\}_{s\in S}\in V$ be the system of simple roots in the standard geometric realization, so the $\{\alpha_s\}$ are linearly independent. The bilinear form $\bullet$ given by $$\alpha_s\bullet\alpha_s = \begin{cases}-\cos\frac{\pi}{m_{s,t}}, & m_{s,t}<\infty\\ -1, & m_{s,t}=\infty\end{cases}$$ has signature $(n-1, 0)$.

It's well-known that ${\rm Rad}(\bullet)$ is one-dimensional, and is spanned by $\delta=\sum_{s}c_s\alpha_s$ where all $c_s>0$, and $\bullet$ is positive-definite on the quotient $U=V/\mathbb{R}\delta$, which can be identified with the hyperplane in the dual space $U^\ast=\langle\cdot,\delta\rangle=0$. $W$ also acts on $U^\ast$ as a reflection group, this gives a homomorphism $W\to{\rm GL}(U^\ast)$, and let $W'$ be its kernel. Prove that $W'$ is non-trivial.

Is there a direct proof of this?

Update: Sorry for the confusion. By 'direct', I mean a general one that doesn't need to check the list of irreducible affine root systems.

Answer 1

When you ask for a direct proof, I'm not sure what I'm allowed to use. But if I am allowed to know how to construct the affine root system from an appropriate finite root system, the fact that $W'$ is non-trivial becomes easy:

If you use the standard construction of an affine root system for your Coxeter group, the root system is the set of vectors $\beta+k\delta$ such that $\beta$ is a root in the finite root system and $k$ is an integer. For two roots with the same $\beta$ and different $k$, the corresponding reflections have the same action on $U^*$.