We initiate the study of affine Deligne–Lusztig varieties with arbitrarily deep level structure for general reductive groups over local fields. We prove that for \({{\,\mathrm{GL}\,}}_n\) and its inner forms, Lusztig’s semi-infinite Deligne–Lusztig construction is isomorphic to an affine Deligne–Lusztig variety at infinite level. We prove that their homology groups give geometric realizations of the local Langlands and Jacquet–Langlands correspondences in the setting that the Weil parameter is induced from a character of an unramified field extension. In particular, we resolve Lusztig’s 1979 conjecture in this setting for minimal admissible characters.

我们开始研究具有任意深度层次结构的仿射Deligne–Lusztig仿射品种，以用于当地田地上的一般还原性群体。我们证明，对于\（{{\，\ mathrm {GL} \，}} _ n \）及其内部形式，Lusztig的半无限Deligne-Lusztig构造在无限级上与仿射Deligne-Lusztig变体同构。我们证明，在Weil参数是由未分叉的场扩展特征引起的情况下，它们的同源性组给出了本地Langlands和Jacquet-Langlands对应关系的几何实现。特别是，我们在这种情况下解决了Lusztig的1979年猜想，以使可允许的字符最少。