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 W yxy^{-1 = x' \} for all elements x, x' of any affine Coxeter group W. The centralizer of x in W is the special case C (x, x). The key structure in our description of the conjugacy class x is the mod-set Modₖ (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 Modₖ (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 Modₖ (w) equals the dimension of Mov (w), and then further investigate type-by-type the surprisingly subtle structure of the Z-module ModW (w). As corollaries, we determine exactly when Modₖ (w) = Mov (w) R^, in which case our closed-form descriptions of conjugacy classes and coconjugation sets are as simple as possible.