What does it mean for a Coxeter system to be of "spherical" type?

Question

In the theorem of the paper Sur les valeurs propres de la transformation de Coxeter the author uses in the main theorem the term "spherical" to refer to a property that Coxeter systems $(W,S)$ can have. What exactly does that term mean?

Answer 1

Let $\Gamma$ be a Coxeter graph and let $(W,S)$ be the Coxeter system of $\Gamma$. We say $\Gamma$ is $\textit{of spherical type}$ if $W_{\Gamma}$ is finite.

Note that if $\Gamma_1,\ldots, \Gamma_{\ell}$ are the connected components of $\Gamma$, then $W=W_{\Gamma_1}\times \cdots \times W_{\Gamma_{\ell}}$. In this case, $\Gamma$ is of spherical type if and only if each of its connected component is of spherical type.

Some properties are:

1. A Coxeter graph $\Gamma$ is of spherical type if and only if the symmetric bilinear form $b:V\times V\rightarrow \mathbb{R}$ is positive definite, where $V$ is a representation space of $W$.
2. The connected spherical type Coxeter graphs are precisely the following:

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Note: the image has been taken from Mike Davis' slides on "Examples of Groups: Coxeter Groups".