COMMENSURATORS OF PARABOLIC SUBGROUPS OF COXETER GROUPS

Let (W, S) be a Coxeter system, and let X be a subset of S. The subgro up of W generated by X is denoted by W-X and is called a parabolic sub group. We give the precise definition of the commensurator of a subgro up in a group. In particular, the commensurator of W-X in W is the sub group of w in W such that wW(X)w(-1) boolean AND 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-X0 . W-X infinity similar or equal to W-X0 x W-X infinity where W-X0 is finite and all the irreducible components of W-X infini ty are infinite. Let Y-infinity be the set of t in S such that m(s,t) = 2 for all s is an element of X(infinity). We prove that the commensu rator of W-X is W-Y infinity . W-X infinity similar or equal to W-y in finity x W-X infinity. 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-infinity.