Non-cuspidality outside the middle degree of l-adic cohomology of the Lubin-Tate tower

TL;DR

This paper proves that, in the l-adic cohomology of the Lubin-Tate tower, only middle degree cohomology contains supercuspidal representations, simplifying previous proofs and avoiding the use of Shimura varieties.

Contribution

It provides a direct, purely local proof that supercuspidal representations only appear in the middle degree cohomology of the Lubin-Tate tower.

Findings

Supercuspidal representations do not appear outside the middle degree.

The proof is purely local and does not involve Shimura varieties.

The result simplifies understanding of the cohomology structure.

Abstract

In this article, we consider the representations of the general linear group over a non-archimedean local field obtained from the vanishing cycle cohomology of the Lubin-Tate tower. We give an easy and direct proof of the fact that no supercuspidal representation appears as a subquotient of such representations unless they are obtained from the cohomology of the middle degree. Our proof is purely local and does not require Shimura varieties.