Affine Deligne-Lusztig varieties of higher level and the local Langlands correspondence for $G{L}_2$

In the present article we define coverings of affine Deligne-Lusztig varieties attached to a connected reductive group over a local field of characteristic $p>0$. In the case of ${\text{GL}}_2$, the unramified part of the local Langlands correspondence is realized in the $\mathrm{\ell }$-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.