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 π is an irreducible admissible representation of GL(n, F ), then, after a finite sequence of cyclic base changes, the image of π 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 π of G(F ) to be one with the property that, after any sequence of cyclic base changes, the image of π contains a supercuspidal member. If F is of positive characteristic then we define π to be pure if the Langlands parameter attached to π by Genestier and Lafforgue is pure in an appropriate sense. We conjecture that no pure supercuspidal representation is incorrigible. We sketch a proof of 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.This paper is an outgrowth of the author's paper arXiv:1609.03491 with G.