A note on element centralizers in finite Coxeter groups

The normalizer N(W)(W(J)) of a standard parabolic subgroup W(J) of a finite Coxeter group W splits over the parabolic subgroup with complement N(J) consisting of certain minimal length coset representatives of W(J) in W. In this note we show that (with the exception of a small number of cases arising from a situation in Coxeter groups of type D(n)) the centralizer C(W)(w) of an element w epsilon W is in a similar way a semidirect product of the centralizer of w in a suitable small parabolic subgroup W(J) with complement isomorphic to the normalizer complement N(J). Then we use this result to give a new short proof of Solomon's Character Formula and discuss its connection to MacMahon's master theorem.