Affine Deligne–Lusztig varieties at infinite level

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