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Stabilizer of a fundamental parabolic face

PDF p. 12: Lemma 3.23, proof, second paragraph

Statement

Let $(W, S)$ be the Coxeter system, $V$ its geometric representation with simple-root basis $(α_{i})_{i \in Γ_{0}}$ , and $Θ = V^{*}$ with its contragredient $W$ -action. The index set may be infinite. For every $J \subseteq Γ_{0}$ and every $φ \in C_{J}$ , the stabilizer ${w \in W : w φ = φ}$ is exactly the standard parabolic subgroup $W_{J}$ .

Statement references

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

Standard parabolic subgroups and parabolic roots

Parabolic coordinate subspaces and fundamental faces

Facts & Assumptions

Given: The Coxeter system and $φ \in C_{J}$ in the statement. Goal: Prove both inclusions between its stabilizer and $W_{J}$ .

[F1]

By Coxeter-system, roots, length and weak-order conventions, every $w \in W$ has a word involving finitely many simple generators, $s_{i} (v) = v - 2 B (v, α_{i}) α_{i}$ , and $(w φ) (v) = φ (w^{- 1} v)$ . By Parabolic coordinate subspaces and fundamental faces, $φ (α_{i}) = 0$ for $i \in J$ and $φ (α_{i}) > 0$ for $i \notin J$ . By Standard parabolic subgroups and parabolic roots, $W_{J}$ is generated by the $s_{i}$ with $i \in J$ .

[F2]

For every finite subset $T \subseteq Γ_{0}$ , Finite-generator standard parabolics and the Bruhat subword property at arbitrary rank identifies the Coxeter group for the restricted matrix on $T$ with the subgroup of $W$ generated by ${s_{i} : i \in T}$ .

[F3]

For a finite-rank Coxeter system, Tits cone geometry with its exact interior boundary states that the stabilizer of a functional in a fundamental face is the subgroup generated by the simple reflections at its zero coordinates.

Proof

1.1F1algebra

For $i \in J$ and $v \in V$ , the reflection formula gives $(s_{i} φ) (v) = φ (s_{i} v) = φ (v) - 2 B (v, α_{i}) φ (α_{i}) = φ (v)$ . Thus each generator of $W_{J}$ fixes $φ$ , and so $W_{J}$ is contained in its stabilizer.

2.1F1F2F3step 1.1∎

Conversely, suppose $w φ = φ$ . Choose a finite $T \subseteq Γ_{0}$ containing all indices in a word for $w$ . The span $V_{T}$ of the corresponding simple roots is invariant under those generators, and their restricted reflection formulas are the canonical geometric representation for the restricted Coxeter matrix. By [F2], this finite-rank Coxeter group embeds as $W_{T}$ . The restriction $φ ∣_{V_{T}}$ is fixed by $w$ , since $w^{- 1} V_{T} = V_{T}$ , and its zero coordinates are exactly $J \cap T$ . By [F3], $w$ belongs to the subgroup generated by ${s_{i} : i \in J \cap T}$ , which is contained in $W_{J}$ . Together with step 1.1 this proves equality.

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Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF p. 12, Lemma 3.23, proof, second paragraph