# Affine Deligne-Lusztig varieties and the geometry of Euclidean reflection groups

Let $G$ be a reductive group such as $SL_n$ 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 $G(F)$.  The associated affine Deligne-Lusztig varieties $X_x(b)$ were introduced by Rapoport.  These are indexed by elements $x$ in $W$ and $b$ in $G(F)$, and are related to many important concepts in algebraic geometry over fields of positive characteristic.  Basic questions about the varieties $X_x(b)$ which have remained largely open include when they are nonempty, and if nonempty, their dimension.  For these questions, it suffices to consider elements $x$ and $b$ both in $W$.  We use techniques inspired by representation theory and geometric group theory to address these questions in the case that $b$ is a translation.  Our approach is constructive and type-free, sheds new light on the reasons for existing results and conjectures, and reveals new patterns.  Since we work only in the standard apartment of the affine building for $G(F)$, which is just the tessellation of Euclidean space induced by the action of the reflection group $W$, our results also hold over the $p-adics$.  We obtain applications to class polynomials of affine Hecke algebras and to reflection length in $W$.  This is joint work with Elizabeth Milicevic (Haverford) and Petra Schwer (Karlsruhe).