Affine Deligne-Lusztig varieties of higher level and the local Langlands correspondence for GL2

In the present article we define coverings of affine Deligne-Lusztig varieties attached to a connected reductive group over a non-archimedean local field. In the case of GL2 and positive characteristic, the unramified cuspidal part of the local Langlands correspondence is realized in the ℓ-adic cohomology of these varieties. We show this by giving a detailed comparison with the realization of local Langlands via cuspidal types by Bushnell-Henniart. All proofs are purely local.