On loop Deligne--Lusztig varieties of Coxeter-type for inner forms of ${\rm GL}_n$

TL;DR

This paper explores loop Deligne--Lusztig varieties for inner forms of GL_n, showing their cohomology realizes many supercuspidal representations and provides a geometric approach to local Langlands correspondence.

Contribution

It simplifies the proof of representability of loop Deligne--Lusztig varieties and demonstrates their cohomology captures key supercuspidal representations for inner forms of GL_n.

Findings

Cohomology realizes many supercuspidal representations.

Provides a geometric realization of local Langlands correspondence.

Simplifies the proof of representability of loop Deligne--Lusztig varieties.

Abstract

For a reductive group $G$ over a local non-archimedean field $K$ one can mimic the construction from the classical Deligne--Lusztig theory by using the loop space functor. We study this construction in special the case that $G$ is an inner form of ${\rm GL}_n$ and the loop Deligne--Lusztig variety is of Coxeter type. After simplifying the proof of its representability, our main result is that its $\ell$-adic cohomology realizes many irreducible supercuspidal representations of $G$, notably almost all among those whose L-parameter factors through an unramified elliptic maximal torus of $G$. This gives a purely local, purely geometric and -- in a sense -- quite explicit way to realize special cases of the local Langlands and Jacquet--Langlands correspondences.