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Affine Deligne-Lusztig varieties of higher level and the local Langlands correspondence for $GL_2$ (1504.00264v1)
Published 1 Apr 2015 in math.AG and math.RT
Abstract: 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 $\GL_2$, the unramified part of the local Langlands correspondence is realized in the $\ell$-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.