Let $F_v$ be a local field. Let $\sigma_v$ be a two-dimensional representation of $Gal(\overline{F}_v:F_v) \rtimes W_{F_v}$. Now, there exists an infinite-dimensional representation $\pi_v$ of $GL_2(F_v)$, whose local $L$-functions and root number coincides with that of $\sigma_v$ .

How and to what extent can we read off the $\pi_v$ from $\sigma_v$?

When is $\pi_v$ a supercuspidal rep, an unramified, or ramified principal series rep, a Steinberg representation, a discrete series representation? What properties of $\sigma_v$ are decisive?

nothavefinite image. – Joël May 2 '13 at 15:05Frobenius semisimple Weil--Deligne representation, which is a representation of the Weil group $W_{F_v}$ which has open kernel when restricted to inertia, together with a nilpotent operator that satisfies a certain commutation relation. If $\sigma_v$ is unramified (including $N = 0$), we get unram. principal series. If $\sigma_v$ is ramified but reducible, and $N \neq 0$, we get ... – Emerton May 10 '13 at 23:20