My naive picture of the local Langlands correspondence for $GL(2,\mathbf{C})$ is this. The Weil group of $\mathbf{C}$ is canonically $\mathbf{C}^\times$. On the Galois side then we're looking at 2-dimensional semisimple representations of $\mathbf{C}^\times$, that is, pairs of continuous group homomorphisms $(\chi_1,\chi_2)$ with $\chi_i:\mathbf{C}^\times\to\mathbf{C}^\times$, modulo $(\chi_1,\chi_2)\sim(\chi_2,\chi_1)$.

On the representation theory side we're looking at irreducible admissible complex representations of $GL(2,\mathbf{C})$ and it's a standard fact that we can build them all from principal series (in the sense described below). Given a pair $(\chi_1,\chi_2)$ as above we can build a 1-dimensional representation of the upper triangular matrices in $GL(2,\mathbf{C})$ and then induce up (normalised induction) to get a principal series representation $I(\chi_1,\chi_2)$ of $GL(2,\mathbf{C})$.

If all the principal series representations were irreducible, and $I(\chi_1,\chi_2)\cong I(\chi_2,\chi_1)$ life would be great: we match up $\chi_1\oplus\chi_2$ with $I(\chi_1,\chi_2)$ and there's the correspondence.

I don't think life is quite so easy though, because there are some reducible principal series. Now the standard trick seems to be that you order the $\chi_i$ by rate of growth of absolute value and then show $I(\chi_1,\chi_2)$ has a unique irreducible quotient $J(\chi_1,\chi_2)$, and match $\chi_1\oplus\chi_2$ with $J(\chi_1,\chi_2)$. I think that this is what the local Langlands correpondence is really supposed to be.

I don't get it. If $I(\chi_1,\chi_2)$ has, say, two Jordan-Hoelder factors (one finite-dimensional say) then we get two representations of $GL(2,\mathbf{C})$ "attached" to $(\chi_1,\chi_2)$ and they're probably not going to be isomorphic, so they had better correspond to two different representations of the Weil group. But all we have is $\chi_1$ and $\chi_2$ and they can't both correspond to $\chi_1\oplus\chi_2$. Is the idea that when this happens, one of them is $I(\chi_3,\chi_4)$ for some different pair of characters? Presumably this is easy to see on the representation theory side but I can't spot the intertwiner. Have I got the picture wrong?

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