Cohomology of affine Deligne–Lusztig varieties for [formula omitted

Abstract

In this paper we study affine Deligne–Lusztig varieties Xw(b) for GL2 and their étale coverings. At first, we compute them explicitly, then we determine associated representations of a certain locally-compact group, the group of rational points of the σ-stabilizer of b, in their étale cohomology. Further, we study these representations by determining morphisms into the irreducible representations of the given group. In particular all cuspidal representations of level 0 of GL2 of a local field and of its inner form, which is the group of units of a quaternion algebra, occur in the cohomology.