ON FINITE FACTORS OF CENTRALIZERS OF PARABOLIC SUBGROUPS IN COXETER GROUPS

It has been known that the centralizer Z(W)(W-I) of a parabolic subgroup W-I of a Coxeter group W is a split extension of a naturally defined reflection subgroup by a subgroup defined by a 2-cell complex Y. In this paper, we study the structure of Z(W)(W-I) further and show that, if I has no irreducible components of type A(n) with 2 <= n < infinity, then every element of finite irreducible components of the inner factor is fixed by a natural action of the fundamental group of Y. This property has an application to the isomorphism problem in Coxeter groups.