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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?

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    $\begingroup$ Marc, I think you don't have the global Langlands formalism right. Automorphic forms for $Gl_2(\mathbb A_F)$ do not correspond to two-dimensional Galois representation as you say, but to representation of a much larger group, conjectural, called the Langlands group $L_F$. This group is not profinite in particular, nor are his local counterparts, and complex representations of those will in general not have finite image. $\endgroup$
    – Joël
    Commented May 2, 2013 at 15:05
  • $\begingroup$ I know. I am not saying that automorphic reps can be mapped to Galois reps, but only that Galois reps inject into automorphic forms. $\endgroup$
    – Marc Palm
    Commented May 2, 2013 at 15:14
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    $\begingroup$ Yes. What is the Weil-Deligne group of a global field ? $\endgroup$
    – Joël
    Commented May 2, 2013 at 15:43
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    $\begingroup$ The Yes of my preceding comment answered the question "Are there still objections?". No, I am not talking only about Maass forms. I believe that the four comments I have written below my answer to that question mathoverflow.net/questions/127157/… may be useful to clarify the relevant point. $\endgroup$
    – Joël
    Commented May 2, 2013 at 15:55
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    $\begingroup$ Dear Marc, Your description of $\sigma_v$ is wrong (at least, if you intended to write the usual thing that appears on the "Galois side" of local Langlands). Namely, what $\sigma_v$ should be is a Frobenius 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 ... $\endgroup$
    – Emerton
    Commented May 10, 2013 at 23:20

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