Affine Deligne--Lusztig varieties with finite Coxeter parts

In this paper, we study affine Deligne–Lusztig varieties $X_w(b)$ when the finite part of the element $w$ in the Iwahori–Weyl group is a partial $\sigma$-Coxeter element. We show that such $w$ is a cordial element and $X_w(b)\mathrm{\ne }\mathrm{\varnothing }$ if and only if $b$ satisfies a certain Hodge–Newton indecomposability condition. The main result of this paper is that for such $w$ and $b$, $X_w(b)$ has a simple geometric structure: the $\sigma$-centralizer of $b$ acts transitively on the set of irreducible components of $X_w(b)$; and each irreducible component is an iterated fibration over a classical Deligne–Lusztig variety of Coxeter type, and the iterated fibers are either ${\mathbb{A}}^1$ or ${\mathbb{G}}_m$.