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In the present article we define coverings of affine Deligne-Lusztigvarieties attached to a connected reductive group over a local field ofcharacteristic $p > 0$. In the case of $\GL_2$, the unramified part of thelocal Langlands correspondence is realized in the $\ell$-adic cohomology ofthese varieties. We show this by giving a detailed comparison with therealization of local Langlands via cuspidal types by Bushnell-Henniart. Allproofs are purely local.

Affine Deligne-Lusztig varieties of higher level and the local Langlands correspondence for $GL_2$