Commensurators of parabolic subgroups of Coxeter groups

Let (W,S) be a Coxeter system, and let X be a subset of S. The subgroup of W generated by X is denoted by WX and is called a parabolic subgroup. We give the precise definition of the commensurator of a subgroup in a group. In particular, the commensurator of WX in W is the subgroup of w in W such that wWXw ∩WX has finite index in both WX and wWXw . The subgroup WX can be decomposed in the form WX = WX0 ·WX∞ ≃ WX0 ×WX∞ where WX0 is finite and all the irreducible components of WX∞ are infinite. Let Y ∞ be the set of t in S such that ms,t = 2 for all s ∈ X. We prove that the commensurator of WX is WY ∞ ·WX∞ ≃ WY ∞ ×WX∞ . In particular, the commensurator of a parabolic subgroup is a parabolic subgroup, and WX is its own commensurator if and only if X 0 = Y .