Deep level Deligne--Lusztig representations of Coxeter type

TL;DR

This paper investigates the cohomology of deep level Deligne--Lusztig varieties of Coxeter type for reductive groups over local fields, leading to new irreducible representations of parahoric subgroups and their induction to supercuspidal representations.

Contribution

It introduces a new approach to constructing irreducible representations of parahoric subgroups via deep level Deligne--Lusztig varieties, extending prior results.

Findings

Constructs new irreducible representations of parahoric subgroups.

Shows these representations induce to supercuspidal representations in the quasi-split case.

Extends previous work on Deligne--Lusztig theory for p-adic groups.

Abstract

In this article we study the cohomology of deep level Deligne--Lusztig varieties of Coxeter type, attached to a reductive group over a local non-archimedean field, which splits over an unramified extension. This allows to construct some new irreducible representations of parahoric subgroups of $p$-adic groups. Moreover, in the quasi-split case we prove that these compactly induce to finite direct sums of irreducible supercuspidal representations of the $p$-adic group. This extends previous results of \cite{DI}, \cite{CI_loopGLn}.