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

TL;DR

This paper constructs coverings of affine Deligne-Lusztig varieties for reductive groups over local fields of characteristic p, and demonstrates the unramified local Langlands correspondence for GL_2 via their l-adic cohomology, connecting geometric and representation-theoretic approaches.

Contribution

It introduces new coverings of affine Deligne-Lusztig varieties and establishes their role in realizing the unramified local Langlands correspondence for GL_2 through cohomological methods.

Findings

Realization of unramified local Langlands for GL_2 via these varieties

Comparison with Bushnell-Henniart's cuspidal types approach

Purely local proofs of the correspondence

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.