PreprintDimensions of affine DeligneLusztig varieties: a new approach via labeled folded alcove walks and root operatorsElizabeth Milićević, Petra Schwer and Anne ThomasAbstractLet \(G\) 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 DeligneLusztig 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 typefree, 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 DeligneLusztig set. We present two immediate consequences of our main results, to class polynomials of affine Hecke algebras and to affine reflection length. This paper is available as a pdf (1164kB) file.
