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P What remains is to show that the chamber complex X is isomorphic to the Coxeter complex Σ(W, S) attached to (W, S). Characterization by foldings and walls 47 It is clear that C¯ is a ‘fundamental domain’ for W on X, that is, any vertex (or simplex) in X can be mapped to a vertex (or simplex) inside C¯ by an element of W . Last, we claim that, for a subset S of S, the stabilizer in W of the face of C of type S is the ‘parabolic subgroup’ S of W . Let x be a face of type S . Certainly all reflections in the facets of type s ∈ S stabilize x.

More specifically, these systems are affine, in a sense only clarified later. The spherical case had been appreciated for at least twenty years before the affine phenomenon was discovered. The simplest affine Coxeter system, which is also the infinite dihedral group, is called A˜1 . ) This is the only case among the families we discuss here that some Coxeter datum m(s, t) takes the value +∞. And among affine Coxeter groups this is the only group recognizable in more elementary terms. The description of A˜n for n > 1 is by generators s1 , .

The associated Coxeter complex Σ = Σ(W, S) is defined to be the simplicial complex associated to the Coxeter poset of (W, S). That is, Σ(W, S) has simplices which are cosets in W of the form w T for a proper (possibly empty) subset T of S, with face relations opposite of subset inclusion in W . Of course, when attempting to define a simplicial complex as a poset, there are conditions to be verified to be sure that we really have a simplicial complex. This is done below. Thus, the maximal simplices are of the form w ∅ = {w} for w ∈ W , and the next-to-maximal simplices are of the form w s = {w, ws} for s ∈ S and w ∈ W.

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