Local Jacquet-Langlands correspondence for regular supercuspidal representations

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 $\zeta$-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.