Transfer relations in essentially tame local Langlands correspondence

TL;DR

This paper connects the automorphic induction character identity with the spectral transfer identity in the essentially tame local Langlands correspondence for GL_n over non-Archimedean fields, clarifying the role of transfer factors and rectifiers.

Contribution

It establishes the equivalence of two key identities in the correspondence and expresses the correspondence via Langlands-Shelstad embeddings, linking rectifiers to transfer factors.

Findings

Automorphic induction and spectral transfer identities are shown to be equal when normalized.

The local Langlands correspondence is expressed using Langlands-Shelstad $ ext{chi}$-data.

Rectifiers are related to transfer factors in the endoscopic transfer.

Abstract

Let $F$ be a non-Archimedean local field and $G$ be the general linear group $\mathrm{GL}_n$ over $F$. Bushnell and Henniart described the essentially tame local Langlands correspondence of $G(F)$ using rectifiers, which are certain characters defined on tamely ramified elliptic maximal tori of $G(F)$. They obtained such result by studying the automorphic induction character identity. We relate this identity to the spectral transfer character identity, based on the theory of twisted endoscopy of Kottwitz, Langlands, and Shelstad. In this article, we establish the following two main results. (i) To show that the automorphic induction character identity is equal to the spectral transfer character identity, when both are normalized by the same Whittaker data. (ii) To express the essentially tame local Langlands correspondence using admissible embeddings constructed by Langlands-Shelstad $\chi$-data, and to relate Bushnell-Henniart's rectifiers to certain transfer factors.