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

Abstract

The cohomology of the Lubin-Tate tower is known to realize the local Langlands correspondence for G L ( n ) GL(n) over a nonarchimedean local field. In this article we make progress toward a purely local proof of this fact. To wit, we find a family of formal schemes V \mathcal {V} such that the generic fiber of V \mathcal {V} is isomorphic to an open subset of Lubin-Tate space at infinite level, and such that the middle cohomology of the special fiber of V \mathcal {V} realizes the local Langlands correspondence for a broad class of supercuspidals (those whose Weil parameters are induced from an unramified degree n n extension). The special fiber of V \mathcal {V} is related to an interesting variety X X , defined over a finite field, which is “maximal” in the sense that the number of rational points of X X is the largest possible among varieties with the same Betti numbers as X X . The variety X X is derived from a certain unipotent algebraic group, in an analogous manner as Deligne-Lusztig varieties are derived from reductive algebraic groups.