Some endoscopic properties of the essentially tame Jacquet-Langlands correspondence

Abstract

Let $F$ be a non-Archimedean local field of characteristic 0 and $G$ be an inner form of the general linear group $G^{\ast }=GL$_n over $F$. We show that the rectifying character appearing in the essentially tame Jacquet-Langlands correspondence of Bushnell and Henniart for $G$ and $G^{\ast }$ can be factorized into a product of some special characters, called zeta-data in this paper, in the theory of endoscopy of Langlands and Shelstad. As a consequence, the essentially tame local Langlands correspondence for $G$ can be described using admissible embeddings of L-tori.