Thirty-Third Annual Victorian Algebra Conference, Sydney, Australia, 2015

Register for: VAC33

Details of talk

TitleAffine Deligne-Lusztig varieties and the geometry of Euclidean reflection groups
PresenterAnne Thomas (The University of Sydney)
Author(s)Dr Elizabeth Milicevic, Dr Petra Schwer and Dr Anne Thomas

[No pdf available]
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.