Affine Deligne-Lusztig varieties at infinite level

TL;DR

This paper explores affine Deligne-Lusztig varieties with deep level structures, establishing isomorphisms with Lusztig's semi-infinite construction for GLn, and connecting their homology to local Langlands and Jacquet--Langlands correspondences.

Contribution

It introduces the study of affine Deligne-Lusztig varieties at infinite level for general reductive groups and proves key isomorphisms and realizations in the GLn case.

Findings

Isomorphism between Lusztig's semi-infinite construction and affine Deligne-Lusztig varieties at infinite level for GLn.

Homology groups realize local Langlands and Jacquet--Langlands correspondences for induced Weil parameters.

Resolution of Lusztig's 1979 conjecture for minimal admissible characters.

Abstract

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 GLn 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.