Local Jacquet-Langlands correspondence for regular supercuspidal representations

Abstract

We prove that Kaletha's local Langlands correspondence for regular supercuspidal representations gives the classical local Jacquet-Langlands correspondence due to Deligne-Kazhdan-Vignéras and Badulescu. As in a former joint paper with Oi, where a similar result is proved for the local Langlands correspondence for the general linear group, the key ingredients in our proof are the work of Bushnell-Henniart explicitly describing the local Jacquet-Langlands correspondence for essentially tame supercuspidal representations and its reinterpretation due to Tam in terms of Langlands-Shelstad's ζ-data. While, under suitable assumptions, this result follows from more general theorems either in the recent work of Fintzen-Kaletha-Spice or that of Chan-Oi, our proof does not require any assumptions.We also complement a few points on the proof in the former paper with Oi.