2018/11/12 by Harris, Michael
#11F66 #22E50 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1811.05050
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 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.