  We define a stratification of Deligne--Lusztig varieties and their parahoric
analogues which we call the Drinfeld stratification. In the setting of inner
forms of GLn, we study the cohomology of these strata and give a complete
description of the unique closed stratum. We state precise conjectures on the
representation-theoretic behavior of the stratification. We expect this
stratification to play a central role in the investigation of geometric
constructions of representations of $p$-adic groups.
