Incorrigible Representations

TL;DR

This paper introduces the concept of incorrigible supercuspidal representations in the context of the local Langlands correspondence, conjectures their non-existence for pure cases, and proves this for certain groups, offering a new proof approach.

Contribution

It defines incorrigible supercuspidal representations and conjectures their non-existence for pure cases, providing proofs for $GL(n)$ and classical groups.

Findings

Proves the non-existence of pure incorrigible supercuspidal representations for $GL(n)$ and classical groups.

Provides a new proof of Henniart's theorem and the local Langlands correspondence for $GL(n)$.

Connects properties of $L$-functions with the structure of supercuspidal representations.

Abstract

As a consequence of his numerical local Langlands correspondence for $GL(n)$, Henniart deduced the following theorem: If $F$ is a nonarchimedean local field and if $\pi$ is an irreducible admissible representation of $GL(n,F)$, then, after a finite sequence of cyclic base changes, the image of $\pi$ contains a vector fixed under an Iwahori subgroup. This result was indispensable in all proofs of the local Langlands correspondence. Scholze later gave a different proof, based on the analysis of nearby cycles in the cohomology of the Lubin-Tate tower. Let $G$ be a reductive group over $F$. Assuming a theory of stable cyclic base change exists for $G$, we define an incorrigible supercuspidal representation $\pi$ of $G(F)$ to be one with the property that, after any sequence of cyclic base changes, the image of $\pi$ contains a supercuspidal member. If F is of positive characteristic then we define $\pi$ to be pure if the Langlands parameter attached to $\pi$ by Genestier and Lafforgue is pure in an appropriate sense. We conjecture that no pure supercuspidal representation is incorrigible. We prove this conjecture for $GL(n)$ and for classical groups, using properties of standard $L$-functions; and we show how this gives rise to a proof of Henniart's theorem and the local Langlands correspondence for $GL(n)$ based on V. Lafforgue's Langlands parametrization, and thus independent of point-counting on Shimura or Drinfel'd modular varieties.