Affine Deligne-Lusztig varieties at infinite level

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.

Geometric Structure of Affine Deligne-Lusztig Varieties for $GL_3$

“…Let R be a k-algebra, and let r ∈ Z >0 . The R[[t]]-submodule R[[t]] 3 ⊂ R((t)) 3 is called the standard lattice and denoted by Λ R . Definition 2.2.…”

“…We denote the set of all lattices in R((t)) 3 by Latt(R), and the set of all 0-special lattices by Latt 0 (R). We also define, for N ≥ 1, subsets…”

“…In the Iwahori case, Chan and Ivanov [3] gave an explicit description of certain Iwahori-level affine Deligne-Lusztig varieties for GL n . Each component of the disjoint decomposition described there is a classical Deligne-Lusztig variety times finite-dimensional affine space, and they point out the similarity between their description and the results in [11].…”