Maximal varieties and the local Langlands correspondence for 𝐺𝐿(𝑛)

Abstract

Abstract. The cohomology of the Lubin-Tate tower is known to realize the local Langlands correspondence for GL(n) over a nonarchimedean local field. In this article we make progress towards a purely local proof of this fact. To wit, we find a family of open affinoid subsets of Lubin-Tate space at infinite level, whose cohomology realizes the local Langlands correspondence for a broad class of supercuspidals (those whose Weil parameters are induced from an unramified degree n extension). A key role is played by a certain variety X, defined over a finite field, which is "maximal" in the sense that the number of rational points of X is the largest possible among varieties with the same Betti numbers as X. The variety X is derived from a certain unipotent algebraic group, in an analogous manner as Deligne-Lusztig varieties are derived from reductive algebraic groups.

The cohomology of semi-infinite Deligne–Lusztig varieties

“…We remark that in all previous work (i.e. the h = 2 work of Boyarchenko-Weinstein [BW16] and the primitive-χ, equal-characteristic work of the author in [C15], [C16]), pinning down the nonvan-…”

“…(i) In [BW16,Section 4.4.1], the unipotent group U n,q over F q n is defined to be the group consisting of formal expressions 1 + a 1 · e 1 + · · · + a n · e n which are multiplied according to the rule e i · a = a q i · e i for all 1 ≤ i ≤ n and…”

“…We now describe the progress on these two conjectures prior to the present work. Under the assumption k = 1, Conjectures 1.2 and 1.3 were proved in the h = 2 case by Boyarchenko and Weinstein in [BW16], where they show that the perfection of X 2 is the special fiber of a particular open affinoid V in the Lubin-Tate tower, and then prove that the cohomology of V realizes the local Langlands and Jacquet-Langlands correspondences by calculating H i c (X 2 , Q ℓ ) [χ]. For h > 2, the results are more sparse.…”

“…This approach has the disadvantage that U n,q h,k does not arise naturally from the set-up and also does not have a mixed characteristic analogue. (Only when h = 2 does the unipotent group scheme work in mixed characteristic, and [BW16] makes use of this.) In this paper we work with a unipotent group scheme U h,k over F q that arises naturally from the set-up and has the feature that U h,k (F q ) is a subquotient of O × D and stabilizes a F q n -subscheme X h ⊂ U h,k via the natural multiplication action.…”

On the perfection of schemes

“…Our aim in this paper is to present a detailed and elementary construction of the inverse perfection of an F p -scheme and discuss some of its properties. The (inverse) perfection functor has played, a continues to play, a significant role in algebraic geometry (see, for example, [Ser1,Ser2,BD,BW,Pep,KL]). We believe that our presentation will be useful to all students and researchers that at some point in their studies will need to consider the (inverse) perfection of an F p -scheme.…”