Abstract

Co-authors: Liz Milicevic (Haverford College), Anne Thomas (University of Sydney)
       
We present combinatorial properties of Coxeter groups and buildings and explain how they can be used to study nonemptiness and dimensions of affine Deligne Lusztig varieties (ADLVs). These varieties are sub-varieties of the affine flag variety of an algebraic group. And their nonemptinedd can be stated in terms of galleries and their retracted images in the associated Bruhat-Tits building. In addition we will talk about the problem of exact computation of reflection length in affine Coxeter groups. Here reflection length means the minimal number of elements needed to write a given element as a product of reflections. For a particular class of elements the reflection length can be determined from the dimension of an ADLV .