Elements with finite Coxeter part in an affine Weyl group

Let W-a be an affine Weyl group and eta : W-a -> W-o be the natural projection to the corresponding finite Weyl group. We say that w is an element of W-a has finite Coxeter part if eta(w) is conjugate to a Coxeter element of W-o. The elements with finite Coxeter part are a union of conjugacy classes of W-a. We show that for each conjugacy class O of W-a with finite Coxeter part there exists a unique maximal proper parabolic subgroup W-J of W-a, such that the set of minimal length elements in O is exactly the set of Coxeter elements in W-J. Similar results hold for twisted conjugacy classes. (C) 2012 Elsevier Inc. All rights reserved.