Some Endoscopic Properties of The Essentially Tame Jacquet-Langlands Correspondence

Let $F$ be a non-Archimedean local field of characteristic 0 and $G$ be an inner form of the general linear group $G^*=GL_n$ 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.

2010 Mathematics Subject Classification: Primary 22E50; Secondary 11S37, 11F70.

Keywords and Phrases: essentially tame Jacquet-Langlands correspondence, inner forms, admissible pairs, zeta-data, endoscopy, admissible embeddings

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