Elements with finite Coxeter part in an affine Weyl group

Abstract

Let W a be an affine Weyl group and η: W a → W 0 be the natural projection to the corresponding finite Weyl group. We say that w∈Wa has finite Coxeter part if η(w) is conjugate to a Coxeter element of W 0. 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. © 2012 Elsevier Inc.