Imaginary cones and limit roots of infinite Coxeter groups

Abstract

Abstract. Let (W, S) be an infinite Coxeter system. To each geometric representation of W is associated a root system. While a root system lives in the positive side of the isotropic cone of its associated bilinear form, an imaginary cone lives in the negative side of the isotropic cone. Precisely on the isotropic cone, between root systems and imaginary cones, lives the set E of limit points of the directions of roots. In this article we study the close relations of the imaginary cone with the set E, which leads to new fundamental results about the structure of geometric representations of infinite Coxeter groups. In particular, we show that the W -action on E is minimal and faithful, and that E and the imaginary cone can be approximated arbitrarily well by sets of limit roots and imaginary cones of universal root subsystems of W , i.e., root systems for Coxeter groups without braid relations (the free object for Coxeter groups). Finally, we discuss open questions as well as the possible relevance of our framework in other areas such as geometric group theory.

Small roots, low elements, and the weak order in Coxeter groups

“…Observe that the condition B(α, γ) > 0 is equivalent to say that γ is on the positive side of the hyperplane H α = ker(v → B(α, v)) and that the condition −1 < B(α, β) < 1 is equivalent to say that the dihedral reflection subgroup generated by s α , s β is finite, which is equivalent to (Rα + Rβ) ∩ Φ is finite, or equivalently, (Rα + Rβ) ∩ Q = {0}; see [18,23] for more details. So the condition (♥) has the following geometric interpretation: for any small root γ ∈ Σ, simple root α ∈ Δ such that γ is on the positive side of the hyperplane H α and maximal dihedral reflection subgroup W of W such that γ ∈ (Φ + W \ Δ W ), one has (Rα + Rβ) ∩ Φ finite (which translates in the language of normalized roots to: the line passing through α and β contains a finite number of normalized roots).…”

“…The ∞-depth has a nice geometric interpretation in the context of normalized roots (see Remarks 2.7 and 2.9). Following [18] we say that β ∈ Φ is visible from …”

“…In order to present examples of rank |S| = 2, 3, 4 easily, it is useful to consider the projective geometric representation of W , as in [23,18]. Since Φ = Φ + Φ − is encoded by the set of positive roots Φ + , we represent Φ by an 'affine cut' Φ: there is an affine hyperplane V 1 in V transverse to Φ + , i.e., for any β ∈ Φ + , the ray R + β intersects V 1 in a unique non-zero point β.…”

“…Then α ≺ β if and only if β is visible from α looking at Q; see[18, Proposition 5.7]. For instance, inFig.…”

“…([18, Figure 7]) Projective representation of a infinite dihedral reflection subgroup W . In green are the two limit points of the root subsystem associated to W together with the action of W .…”