Elements with finite Coxeter part in an affine Weyl group

摘要

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.