Stratifications of affine Deligne-Lusztig varieties

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.