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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