  Let W be an Iwahori-Weyl group of a connected reductive group G over a
non-archimedean local field. I prove that if w is an element of W that does not
act on the corresponding apartment of G by a translation then one can apply to
w a sequence of conjugations by simple reflections, each of which is
length-preserving, resulting in an element w' for which there exists a simple
reflection s such that l(sw's)>l(w'). Even for affine Weyl groups, a special
case of Iwahori-Weyl groups and also an important subclass of Coxeter groups,
this is a new fact about conjugacy classes. Further, there are implications for
Iwahori-Hecke algebras H of G: one can use this fact to give dimension bounds
on the "length-filtration" of the center Z(H), which can in turn be used to
prove that suitable linearly-independent subsets of Z(H) are a basis.
