Local tame lifting for GL(n) III: explicit base change and Jacquet-Langlands correspondence

Abstract

Let F be a finite extension of ℚ_p with p ≠2, and D a central F-division algebra of dimension p^2m. Let π be an irreducible supercuspidal representation of GL_pm(F). The Jacquet-Langlands correspondence associates to π an irreducible smooth representation π_D of D^x, determined up to isomorphism by a character relation. Using a variant of the description of irreducible supercuspidal representations of GL_n(F) as induced representations, due to Bushnell and Kutzko, along with a parallel description for D^x due to Broussous, we give an explicit realization of the correspondence π ↦ π_D in the case where π is totally ramified. This is a step towards our main result. Let K∣F be a finite unramified extension, and π a totally ramified supercuspidal representation of GL_pm(F). Base change, in the sense of Arthur and Clozel, gives a totally ramified supercuspidal representation b_K∣F π of GL_pm(K). In earlier work, the authors gave an explicit definition of a representation ℓ_K∣F π and showed that ℓ_K∣F π = b_K∣F π when p does not divide the degree of K∣F. We complete this by showing that ℓ_K∣F π = b_K∣F π for all K∣F. The proof relies on evaluating the twisted character of ℓ_K∣F π in terms of the character of π_D and then using the explicit Jacquet-Langlands correspondence. Many of the central arguments remain valid when F is a non-Archimedean local field of odd positive characteristic.