On Certain Admissible Embeddings of L-groups

TL;DR

This paper explores the construction of admissible embeddings of L-groups for tori in GL_n over local fields, linking these embeddings to induced Weil group representations and offering new insights into the local Langlands correspondence.

Contribution

It introduces a method to construct admissible L-group embeddings using Langlands-Shelstad $ ext{chi}$-data, connecting them to induced Weil group representations and reinterpretations of existing correspondence results.

Findings

Constructed admissible embeddings of L-groups using $ ext{chi}$-data.

Established a link between embeddings and induced Weil group representations.

Provided a new perspective on the essentially tame local Langlands correspondence.

Abstract

Let $F$ be a local field and $E/F$ be a separable extension of degree $n$. Regard $T=\text{Res}_{E/F} \mathbb{G}_m$ as an elliptic maximal torus of $G=\mathrm{GL}_n$. We can construct an admissible embedding of L-groups ${}^LT\hookrightarrow {}^LG$ using Langlands-Shelstad $\chi$-data. Such embedding gives rise to an induced representation of the Weil group $W_F$ of $F$ from a character of $W_E$. The relation between induced representations and admissible embeddings provides a different interpretation of the work of Bushnell-Henniart on the essentially tame local Langlands correspondence.