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 $GL(n)$ 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 $\mathcal {V}$ such that the generic fiber of $\mathcal {V}$ is isomorphic to an open subset of Lubin-Tate space at infinite level, and such that the middle cohomology of the special fiber of $\mathcal {V}$ realizes the local Langlands correspondence for a broad class of supercuspidals (those whose Weil parameters are induced from an unramified degree $n$ extension). The special fiber of $\mathcal {V}$ is related to an interesting 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.