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Faithful normaliser-quotient action on the relative cone

PDF p. 12: Lemma 3.23

Statement

Let $(W, S)$ be a Coxeter system with finite vertex set $Γ_{0}$ , and let $V$ be its canonical real representation with simple-root basis $(α_{i})_{i \in Γ_{0}}$ and dual space $Θ = V^{*}$ , equipped with the contragredient action. Fix $J \subseteq Γ_{0}$ . Write $W_{J} = ⟨ s_{j} : j \in J ⟩$ , $Θ_{J} = {f \in Θ : f (α_{j}) = 0 for all j \in J}$ , and $C_{L} = {f \in Θ_{L} : f (α_{i}) > 0 for all i \notin L}$ for $L \subseteq Γ_{0}$ . Let $C o n e$ be the union of the $W$ -translates of the closed fundamental chamber, and put $C o n e (J) = C o n e \cap Θ_{J}$ .

The normaliser ${Norm}_{W} (W_{J}) = {g \in W : g W_{J} g^{- 1} = W_{J}}$ preserves $Θ_{J}$ , and the kernel of its action on $Θ_{J}$ is exactly $W_{J}$ . Consequently its quotient $N (W, J) = {Norm}_{W} (W_{J}) / W_{J}$ acts faithfully by linear maps on $Θ_{J}$ . This action preserves $C o n e (J)$ and its stratification into the translates $x C_{L} \subseteq Θ_{J}$ : for every such stratum and every lift $g \in {Norm}_{W} (W_{J})$ of $\overset{ˉ}{g} \in N (W, J)$ , one has $\overset{ˉ}{g} (x C_{L}) = (g x) C_{L}$ .

Statement references

Coxeter-system, roots, length and weak-order conventions

Standard parabolic subgroups and parabolic roots

Parabolic coordinate subspaces and fundamental faces

Normalisers and their quotients

Tits cone intersections, relative roots and arrangements

Convexity, stratification and faces of the relative cone

Facts & Assumptions

[F1]

For each simple reflection, $s_{i} (v) = v - 2 B (v, α_{i}) α_{i}$ and $B (α_{i}, α_{i}) = 1$ ; the dual action satisfies $(w f) (v) = f (w^{- 1} v)$ . Each $s_{i}$ is an involution. Coxeter-system, roots, length and weak-order conventions

[F2]

The subgroup $W_{J}$ is generated by the simple reflections $s_{j}$ for $j \in J$ , and $C_{J}$ consists of the functionals vanishing on those simple roots and positive on the others. Standard parabolic subgroups and parabolic roots Parabolic coordinate subspaces and fundamental faces

[F3]

The stabilizer in $W$ of every point of $C_{J}$ is $W_{J}$ . Stabilizer of a fundamental parabolic face

[F4]

The Tits cone is $W$ -invariant. Basic Tits-cone properties

[F5]

The relative cone is the disjoint union of the translates $x C_{L}$ that are contained in $Θ_{J}$ . Convexity, stratification and faces of the relative cone

[F6]

$W_{J}$ is normal in its normaliser, so the quotient by $W_{J}$ exists. Normalizer quotients, kernels and conditional surjectivity

Proof

Given: The finite-rank Coxeter system, its geometric and dual actions, and the subset $J$ in the Statement.

Proof technique: direct.

Goal: Prove the invariance and kernel assertions, and the induced action on the relative cone and its strata.

1.1F1F2given

For $f \in Θ$ and $j \in J$ , [F1] gives $(s_{j} f) (v) = f (v) - 2 B (v, α_{j}) f (α_{j})$ . Thus $s_{j}$ fixes $f$ exactly when $f (α_{j}) = 0$ : the forward direction follows by evaluating at $α_{j}$ , and the reverse direction follows from the displayed formula. By [F2], the common fixed subspace of $W_{J}$ is therefore exactly $Θ_{J}$ .

2.1F6step 1.1given

Let $g$ normalize $W_{J}$ and let $f \in Θ_{J}$ . For every $h \in W_{J}$ , the element $g^{- 1} h g$ lies in $W_{J}$ , so $h (g f) = g ((g^{- 1} h g) f) = g f$ by step 1.1. Hence $g f \in Θ_{J}$ . The same argument applied to $g^{- 1}$ gives $g Θ_{J} = Θ_{J}$ .

2.2F2F3step 1.1given

Step 1.1 shows that every element of $W_{J}$ acts trivially on $Θ_{J}$ . Conversely, if an element of the normaliser acts trivially on $Θ_{J}$ , it fixes the functional $f_{J}$ with $f_{J} (α_{j}) = 0$ for $j \in J$ and $f_{J} (α_{i}) = 1$ for $i \notin J$ . The simple roots form a basis, so this functional exists and belongs to $C_{J}$ . By [F3] its stabilizer is $W_{J}$ , proving the reverse inclusion for the kernel.

3.1F1F6step 2.1step 2.2

By [F6] and step 2.2, the restricted action descends to the quotient: define $(g W_{J}) f = g f$ . This is independent of the representative because $W_{J}$ fixes every point of $Θ_{J}$ . Its kernel is trivial by step 2.2, and its maps are linear because the original dual action is linear. Thus the quotient action is faithful.

4.1F4F5step 2.1step 3.1given∎

By [F4] and step 2.1, each normaliser element preserves both $C o n e$ and $Θ_{J}$ , hence preserves their intersection $C o n e (J)$ . If $x C_{L}$ is a stratum in [F5], its image under a lift $g$ is $(g x) C_{L}$ by the action law and is contained in $Θ_{J}$ . It is therefore another stratum in [F5]. The formula is independent of the lift by step 3.1; applying the same argument to $g^{- 1}$ shows that the strata are permuted. This proves all assertions.

Source notes

The cited article attributes the statement. The proof uses the fixed subspace of $W_{J}$ and the listed suppliers; it does not use the article lemma as a premise.

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 p. 12, Lemma 3.23