Let $F$ be a nonarchimedian local field. Since the Weil group $W\_F$ is a dense subgroup of $G\_F=Gal(\bar{F}/F)$, it's clear that restriction gives an injection $Irr(G\_F)\rightarrow Irr(W\_F)$ of irreducible (complex) representations.


In some notes of Prasad and Raghuram (found here: <http://www.math.tifr.res.in/~dprasad/ictp2.pdf>), they assert that the representations are in fact the same "perhaps after a twist." This fact is crucial for their argument that reduces establishing LLC for $GL\_n$ to the case of supercuspidal representations. But I'm not sure I understand what it means. 


So my question is: how does one take a representation of $W\_F$ and twist it in such a way that it can be extended to a representation of $G\_F$?



Extending a representation from the Weil group to the Galois group
Pick $V \in Irr(W\_F)$. Due to Schur's lemma and the fact that inertia in $W\_F$ acts through a finite quotient, some power of (a fixed choice of) Frobenius acts as a scalar on $V$. Take an appropriate root of that scalar to make up an unramified character $\chi$ of $W\_F$ such that the same power of Frobenius acts trivially on $U := V \otimes \chi^{-1}$. I claim that $U \in Irr(G\_F)$, i.e., that $W\_F$ acts through a finite quotient on $U$. Indeed, that quotient is generated by a finite quotient of the inertia and the finite order image of the chosen Frobenius.


