  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.

Representation Theory