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Tits cone geometry with its exact interior boundary

Statement

Let $(W, S)$ be a Coxeter system with $S$ finite, in its canonical real reflection representation on $V = R^{S}$ , with basis $(e_{s})_{s \in S}$ , root system $Φ$ and positive roots $Φ_{+}$ . Let $Θ = V^{*}$ carry the contragredient action. Put $\overline{C} = {f \in Θ : f (e_{s}) \geq 0 for all s \in S}$ and $U = ⋃_{w \in W} w \overline{C}$ . For $I \subseteq S$ , let $C_{I}$ be the set of points of $\overline{C}$ whose zero coordinates are exactly those indexed by $I$ , and let $W_{I}$ be the subgroup generated by $I$ . The root hyperplane of $α \in Φ$ is $H_{α} = {f : f (α) = 0}$ ; interiors and neighborhoods use the ordinary finite-dimensional topology on $Θ$ .

A functional $f$ belongs to $U$ exactly when ${α \in Φ_{+} : f (α) < 0}$ is finite. The cone $U$ is convex, and $\overline{C}$ meets each $W$ -orbit in $U$ exactly once. The stabilizer of $f \in C_{I}$ is $W_{I}$ . A point of $C_{I}$ belongs to $U^{\circ}$ exactly when $W_{I}$ is finite; the family of distinct root hyperplanes is locally finite on $U^{\circ}$ . Moreover $U = Θ$ exactly when $W$ is finite. In general local finiteness of chambers does not extend to boundary points: if $W$ is infinite, every neighborhood of $0$ meets infinitely many distinct chambers $w \overline{C}$ .

Statement references

The Tits cone, its interior, and the negative-root set of a functional

The dual action, chambers, faces, and root hyperplanes

Interior, closure, boundary, limit point, isolated point and dense subset of a metric space

Facts & Assumptions

[F1]

The definition of the Tits cone gives $U = ⋃_{w \in W} w \overline{C}$ and $0 \in U$ ; its interior is taken in the ordinary finite-dimensional topology. The dual action, coordinate faces and root hyperplanes are as defined in The Tits cone, its interior, and the negative-root set of a functional and The dual action, chambers, faces, and root hyperplanes.

[F2]

The finite-negativity criterion holds, and $U$ is closed under convex combinations and nonnegative scalar multiplication. (The finite-negativity criterion, the reduction step, and convexity of the Tits cone, clauses (1) and (4).)

[F3]

Each orbit in $U$ meets $\overline{C}$ once; the stabilizer of $f \in \overline{C}$ is $W_{{s : f (e_{s}) = 0}}$ ; and the walls of the chamber system are exactly the root hyperplanes. (Chamber collisions, point stabilizers, and the intersection rule, clauses (1), (4), and (6).)

[F4]

For $f \in \overline{C}$ , membership in $U^{\circ}$ is equivalent to finiteness of its coordinate-zero subgroup. Every point of $U^{\circ}$ has a neighborhood meeting finitely many walls. Also $0 \in U^{\circ}$ exactly when $W$ is finite, and if $W$ is infinite every neighborhood of $0$ meets infinitely many distinct chambers. (The interior of the Tits cone, finite parabolic stabilizers, and local finiteness, clauses (1), (3), (4), and (5).)

[F5]

In the metric topology on $Θ$ specified in [F1], the closure $cl (A)$ consists of points whose every open ball meets $A$ , and $\partial A = cl (A) ∖ int (A)$ . In particular $A \subseteq cl (A)$ for every $A \subseteq Θ$ . (Interior, closure, boundary, limit point, isolated point and dense subset of a metric space, Definition.)

Proof

Given: The finite-rank Coxeter system and notation in the statement. Goal: Establish the listed properties of $U$ .

1.1F2F3

The finite-negativity criterion and convexity are precisely the first and fourth assertions of [F2]. The orbit and stabilizer assertions follow from [F3], since the zero-coordinate set of a point of $C_{I}$ is $I$ .

1.2F3F4

The interior criterion for $C_{I}$ follows from [F4] and the same identification of the zero-coordinate set. By [F4], each interior point has a neighborhood meeting finitely many walls; by [F3], the distinct walls are exactly the distinct root hyperplanes. Hence those hyperplanes are locally finite on $U^{\circ}$ .

1.3F2F4algebra

If $U = Θ$ , then $0 \in U^{\circ}$ , so $W$ is finite by [F4]. Conversely, if $W$ is finite, [F4] gives an open neighborhood of $0$ contained in $U$ . For any $f \in Θ$ , a sufficiently small positive multiple $t f$ lies in that neighborhood; closure under nonnegative scalar multiplication then gives $f = t^{- 1} (t f) \in U$ . Thus $U = Θ$ .

2.1F1F4F5step 1.3∎

If $W$ is infinite, [F4] says that every neighborhood of $0$ meets infinitely many distinct chambers. Here $0 \in U \subseteq cl (U)$ by [F1] and [F5], while $0 \notin U^{\circ}$ by [F4]. Hence $0 \in cl (U) ∖ U^{\circ} = \partial U$ by [F5], so $0$ is a boundary point of $U$ . This proves the stated boundary qualification and completes the proof.

Depends on

Used by · 13

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