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Let $F$ be a $p$-adic field, $G = \operatorname{Gal}(\overline{F}/F)$ and $W$ the Weil group of $F$. The inclusion map $W \subset G$ is continuous with dense image, so $\rho \mapsto \rho|_W$ defines an injection of continuous irreducible $n$-dimensional complex representations of $G$ into those of $W$.

A continuous representation $\rho: G \rightarrow \operatorname{GL}_n(\mathbb C)$ takes only finitely many values; each $\rho(x)$ is diagonalizable with eigenvalues on the unit circle. This need not be the case if we replace $G$ by $W$.

The local Langlands correspondence attaches irreducible supercuspidal representations of $\operatorname{GL}_n(F)$ to continuous irreducible representations of $W$.

Is there a known description of those irreducible supercuspidal representations $\pi$ of $\operatorname{GL}_n(F)$ which correspond to representations of $G$ (rather than $W$)?

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    $\begingroup$ If the supercuspidal $\pi$ corresponds to the WD irrep $\rho$ under the LLC (normalized as in the book of Harris-Taylor), then we have $\omega_\pi \circ Art_F^{-1} = \det\rho$ ($Art_F$ is the local Artin map and $\omega_\pi$ is the central character of $\pi$). By the proposition in Section 28.6 of Bushnell-Henniart's book, $\rho$ extends to the full Galois group if and only if $\det\rho$ has finite order. So you get supercuspidals whose central character is of finite order. $\endgroup$ – krl May 23 '18 at 22:52

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