Incorrigible 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 LubinTate 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 pointcounting on Shimura or Drinfel'd modular varieties.
 Publication:

arXiv eprints
 Pub Date:
 November 2018
 arXiv:
 arXiv:1811.05050
 Bibcode:
 2018arXiv181105050H
 Keywords:

 Mathematics  Number Theory;
 11F66;
 22E50
 EPrint:
 This paper is an outgrowth of the author's paper arXiv:1609.03491 with G. B\"ockle, S. Khare, and J. Thorne. The second version corrects misprints and incorporates suggestions of J.L. Waldspurger