The geometry of conjugation in affine Coxeter groups

We develop new and precise geometric descriptions of the conjugacy class $[x]$ and coconjugation set C(x,x') = \ y ∈ W̄ yxy⁻¹ = x' \ for all elements $x,x'$ of any affine Coxeter group W̄. The centralizer of x in W̄ is the special case C(x,x). The key structure in our description of the conjugacy class $[x]$ is the mod-set ModW̄(w) = (w-I)R^, where~w is the finite part of x and R^ is the coroot lattice. The coconjugation set C(x,x') is then described by ModW̄(w') together with the fix-set of $w'$, where $w'$ is the finite part of $x'$. For any element w of the associated finite Weyl group W, the mod-set of w is contained in the classical move-set Mov(w) = Im(w - I). We prove that the rank of ModW̄(w) equals the dimension of Mov(w), and then further investigate type-by-type the surprisingly subtle structure of the Z-module ModW̄(w). As corollaries, we determine exactly when ModW̄(w) = Mov(w) ∩ R^, in which case our closed-form descriptions of conjugacy classes and coconjugation sets are as simple as possible.