  Motivated by the problem of giving an explicit description of the basic locus
in the reduction of Shimura varieties, G\"{o}rtz, He and Nie studied the cases
where the basic affine Deligne-Lusztig variety, which serves as its
group-theoretic model, is a union of classical Deligne-Lusztig varieties
associated to Coxeter elements. In this paper, we study a natural
generalization of this stratification in the case of $GL_n$.
