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 $W$)?