A fundamental tool in analyzing the special fiber of Shimura varieties is the Newton stratification, indexed by the set $B(G, \mu)$ of neutral acceptable Frobenius-twisted conjugacy classes. In several key cases, the unique closed Newton stratum - a generalization of the supersingular locus in the Siegel case - admits an explicit description as a union of Deligne-Lusztig varieties. Such descriptions have played a pivotal role in applications to the Kudla-Rapoport program, the Arithmetic Fundamental Lemma, and certain cases of the Tate conjecture. Note that this closed stratum corresponds to the unique smallest element of $B(G, \mu)$, and by Rapoport-Zink uniformization, each Newton stratum is modeled by a corresponding (closed) affine Deligne-Lusztig variety $X(\mu, b)$, a group-theoretic construction reflecting the local structure.

In this talk, we consider the opposite extreme: the maximal element $b_{\mu,\max}$ of $B(G, \mu)$. I will present a dimension formula for the affine Deligne–Lusztig variety $X(\mu, b_{\mu,\max})$ and discuss the structure of its irreducible components. In particular, we will see certain naturally occurring iterated fibrations over Deligne–Lusztig varieties among these components. As an application of our work, we obtain a new proof of the classification of when $X(\mu,b)$ has dimension zero in full generality.