Dimensions of affine Deligne-Lusztig varieties: a new approach via labeled folded alcove walks and root operators

Elizabeth Milićević, Petra Schwer and Anne Thomas

Abstract

Let $G$ be a reductive group over the field $F=k((t))$, where $k$ is an algebraic closure of a finite field, and let $W$ be the affine Weyl group of $G$. The associated affine Deligne-Lusztig varieties $X_x(b)$, which are indexed by elements $b$ in $G(F)$ and $x$ in $W$, were introduced by Rapoport. Basic questions about the varieties $X_x(b)$ which have remained largely open include when they are nonempty, and if nonempty, their dimension. We use techniques inspired by geometric group theory and representation theory to address these questions in the case that $b$ is a pure translation, and so prove much of a sharpened version of a conjecture of Görtz, Haines, Kottwitz, and Reuman. Our approach is constructive and type-free, sheds new light on the reasons for existing results in the case that b is basic, and reveals new patterns. Since we work only in the standard apartment of the building for $G(F)$, our results also hold in the $p$-adic context, where we formulate a definition of the dimension of a $p$-adic Deligne-Lusztig set. We present two immediate consequences of our main results, to class polynomials of affine Hecke algebras and to affine reflection length.