Affine Deligne–Lusztig varieties with finite Coxeter parts

Abstract

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 $X_{w(b)}\ne \varnothing$ if and only if $b$ satisfies a certain Hodge–Newton indecomposability condition. Our main result 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 ${\mathbb{𝔾}}_m$.