On pointwise conjugacy of distinguished coset representatives in Coxeter groups

Let (W, S) be a Coxeter system. For a standard parabolic. subgroup W-K,

K subset of or equal to S let D-K be the set of distinguished coset

representatives, i.e. representatives of cosets W(K)w of minimal

Coxeter length. If L = K-c subset of or equal to S with c is an element

of W, then D-K and D-L = c(-1) D-K are in general not conjugate as

sets. However it is shown that if WK is finite, they are conjugate

'pointwise', i.e. there is a bijection theta : D-K --> D-L such that

theta(d) = d(wc) for some w is an element of W-K depending on d is an

element of D-K. In particular for each conjugacy class C of W the

cardinalities # (D-K boolean AND C) and # (D-L boolean AND C) are the

same. The case of infinite standard parabolic subgroups is also

discussed and a corresponding result is proved.