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Full Description
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 (extended) 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. The authors use techniques inspired by geometric group theory and combinatorial 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 Gortz, Haines, Kottwitz, and Reuman.
The authors' 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 they work only in the standard apartment of the building for $G(F)$, their results also hold in the $p$-adic context, where they formulate a definition of the dimension of a $p$-adic Deligne-Lusztig set. The authors present two immediate applications of their main results, to class polynomials of affine Hecke algebras and to affine reflection length.
Contents
Introduction
Preliminaries on Weyl groups, affine buildings, and related notions
Labelings and orientations, galleries, and alcove walks
Dimensions of galleries and root operators
Affine Deligne-Lusztig varieties and folded galleries
Explicit constructions of positively folded galleries
The varieties $X_x(1)$ in the shrunken dominant Weyl chamber
The varieties $X_x(1)$ and $X_x(b)$
Conjugating to other Weyl chambers
Diagram automorphisms
Applications to affine Hecke algebras and affine reflection length
Bibliography.
