On Certain Admissible Embeddings of L-groups

Let $F$ be a local field and $E\mathrm{/}F$ be a separable extension of degree $n$. Regard $T={\text{Res}}_{E\mathrm{/}F}{\mathbb{G}}_m$ as an elliptic maximal torus of $G={\mathrm{G}\mathrm{L}}_n$. We can construct an admissible embedding of L-groups ${}^{L}T\hookrightarrow {}^{L}G$ 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.