Some endoscopic properties of the essentially tame Jacquet-Langlands correspondence

TL;DR

This paper demonstrates that the rectifying character in the essentially tame Jacquet-Langlands correspondence can be factorized into special zeta-data characters, enabling a description via admissible embeddings of L-tori.

Contribution

It introduces a factorization of the rectifying character into zeta-data, linking endoscopy theory with the local Langlands correspondence for inner forms of GL_n.

Findings

Factorization of the rectifying character into zeta-data.

Connection between endoscopy theory and local Langlands correspondence.

Description of the correspondence using admissible embeddings of L-tori.

Abstract

Let $F$ be a a non-Archimedean local field of characteristic 0 and $G$ be an inner form of the general linear group $G^*=\mathrm{GL}_n$ over $F$. We show that the rectifying character appearing in the essentially tame Jacquet-Langlands correspondence of Bushnell and Henniart for $G$ and $G^*$ 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.