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Basic Tits-cone properties

PDF p. 5: §2.2; PDF p. 7: §3.1; PDF p. 8: opening of Lemma 3.5 argument

Statement

Let $(W, S)$ be a Coxeter system with $S$ finite, with the canonical real reflection representation, its dual space $Θ$ , the contragredient $W$ -action, the closed fundamental chamber $\overline{C}$ , the Tits cone $C o n e = ⋃_{w \in W} w \overline{C}$ , and the set $H$ of distinct root hyperplanes as in Reflecting hyperplanes, fundamental chamber and Tits cone. Use the finite-dimensional topology specified there.

The Tits cone is a $W$ -invariant convex cone. The chamber $\overline{C}$ is a fundamental domain in the sense that every $W$ -orbit in $C o n e$ meets it in exactly one point. The arrangement $({C o n e}^{\circ}, H)$ is locally finite: every point of ${C o n e}^{\circ}$ has a neighbourhood in $Θ$ meeting only finitely many members of $H$ . Finally, $W$ is finite if and only if $C o n e = Θ$ .

Statement references

Reflecting hyperplanes, fundamental chamber and Tits cone

Tits cone geometry with its exact interior boundary

Facts & Assumptions

[F1]

The contragredient maps form a $W$ -action and $C o n e = ⋃_{w \in W} w \overline{C}$ . The hyperplanes in $H$ are the distinct root hyperplanes. Reflecting hyperplanes, fundamental chamber and Tits cone

[F2]

For this finite-rank representation, the union $U = ⋃_{w \in W} w \overline{C}$ is a convex cone, every orbit in $U$ meets $\overline{C}$ exactly once, the distinct root hyperplanes are locally finite on $U^{\circ}$ , and $U = Θ$ exactly when $W$ is finite. Tits cone geometry with its exact interior boundary

Proof

Given: The finite-rank Coxeter system and the geometric notation in the Statement.

Proof technique: direct.

Goal: Prove the four assertions about the Tits cone, its fundamental domain, local finiteness and finiteness of $W$ .

1.1F1F2given

By [F1], for any $g \in W$ the action law gives $g C o n e = ⋃_{w \in W} g w \overline{C} = ⋃_{v \in W} v \overline{C} = C o n e$ , since multiplication by $g$ permutes $W$ . By [F2], the same union is a convex cone and every orbit in it meets $\overline{C}$ exactly once. This proves invariance, convexity and the stated fundamental-domain property.

2.1F1F2step 1.1∎

The definitions in [F1] identify $C o n e$ with the set $U$ in [F2] and $H$ with its distinct root hyperplanes. Thus the local-finiteness assertion in [F2] gives the stated neighbourhood condition on ${C o n e}^{\circ}$ , and its final assertion gives $W$ finite if and only if $C o n e = Θ$ . Together with 1.1 this proves the proposition.

Depends on

Used by · 8

Dependency tree · 2 levels

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Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF pp. 5, 7, 8; §2.2; §3.1; opening of Lemma 3.5 argument