Affine Deligne–Lusztig varieties with finiteCoxeter parts

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 -Coxeter element.We show that such w is a cordial element and X w(b) = 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 -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 1 or m .1. Introduction 1681 2. Preliminaries 1684 3. Class polynomials and reduction trees 1690 4. Cordiality and the set B(G) w 1694 5. Analyzing the reduction paths 1698 6.Some combinatorial identities 1701 7. The general case 1709