Commensurators of parabolic subgroups of Coxeter groups

Abstract:

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 $W_{X}$ 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 $W_{X}$ in $W$ is the subgroup of $w$ in $W$ such that $wW_{X}w^{-1}\cap W_{X}$ has finite index in both $W_{X}$ and $wW_{X}w^{-1}$. The subgroup $W_{X}$ can be decomposed in the form $W_{X} = W_{X^{0}} \cdot W_{X^{\infty }} \simeq W_{X^{0}} \times W_{X^{\infty }}$ where $W_{X^{0}}$ is finite and all the irreducible components of $W_{X^{\infty }}$ are infinite. Let $Y^{\infty }$ be the set of $t$ in $S$ such that $m_{s,t}=2$ for all $s\in X^{\infty }$. We prove that the commensurator of $W_{X}$ is $W_{Y^{\infty }} \cdot W_{X^{\infty }} \simeq W_{Y^{\infty }} \times W_{X^{\infty }}$. In particular, the commensurator of a parabolic subgroup is a parabolic subgroup, and $W_{X}$ is its own commensurator if and only if $X^{0}= Y^{\infty }$.