Affine Deligne–Lusztig varieties with finite Coxeter parts

Abstract

We study affine Deligne–Lusztig varieties Xw(b) when the finite part of the element w in the Iwahori–Weyl group is a partial σ-Coxeter element. We show that such w is a cordial element and Xw(b) ̸= ∅ if and only if b satisfies a certain Hodge–Newton indecomposability condition. Our main result is that for such w and b, Xw(b) has a simple geometric structure: the σ-centralizer of b acts transitively on the set of irreducible components of Xw(b); and each irreducible component is an iterated fibration over a classical Deligne–Lusztig variety of Coxeter type, and the iterated fibers are either A1 or Gm.