Abstract : We present a new approach to affine Deligne Lusztig varieties which allows us to study the so called "non-basic" case in a type free manner. The central idea is to translate the question of non-emptiness and the computation of the dimensions of these varieties into geometric questions in the Bruhat-Tits building. All boils down to understand existence of certain positively folded galleries in affine Coxeter complexes. To do so, we explicitly construct such galleries and use, among other techniques, the root operators introduced by Gaussent and Littelmann to manipulate them.

Keywords : affine Deligne Lusztig varieties; Coxeter groups; Combinatorics; folded galleries