Let $F$ be a number field and let $\sigma : G_{F} = \mathrm{Gal}(\overline{F}/F) \to \mathrm{GL}_{2}(\mathbb{C})$ be a continuous representation of Galois group. Since $G_F$ is compact, the image of $\sigma$ in $\mathrm{PGL}_{2}(\mathbb{C})$ is finite, and the classification for the subgroups is known by Klein. In Gelbart's article on Langlands-Tunnell theorem, it saids that the same thing holds for the representations of Weil group $W_{F}$ (it has same classification on projective image). However, since $W_F$ is not compact, I can't show that the image is finite. Could anyone help?
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By $W_F$, do you want $F$ to be local or global? Even in the local case, you presumably need some (Frobenius-)semisimplicity condition. Otherwise, take the representation of $\mathbb Z$ that sends $1\mapsto\begin{pmatrix}1&1\0&1\end{pmatrix}$ and pull it back to a representation of $W_F$. – Mathmo123 Jul 21 '21 at 08:56
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@Mathmo123 global (number) field – Seewoo Lee Jul 21 '21 at 09:20