  Affine Deligne-Lusztig varieties are analogues of Deligne-Lusztig varieties
in the context of affine flag varieties and affine Grassmannians. They are
closely related to moduli spaces of $p$-divisible groups in positive
characteristic, and thus to arithmetic properties of Shimura varieties. We
compare stratifications of affine Deligne-Lusztig varieties attached to a basic
element $b$. In particular, we show that the stratification defined by Chen and
Viehmann using the relative position to elements of the group $J_b$, the
$\sigma$-centralizer of $b$, coincides with the Bruhat-Tits stratification in
all cases of Coxeter type, as defined by X. He and the author.
