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--Vigneras 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.