Details of talk
|Title||Affine Deligne-Lusztig varieties and the geometry of Euclidean reflection groups|
|Presenter||Anne Thomas (The University of Sydney)|
|Author(s)||Dr Elizabeth Milicevic, Dr Petra Schwer and Dr Anne Thomas|
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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) are indexed by elements x in W and b in G(F), and 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. 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 an application to reflection length in W.