The local Langlands conjecture, by wiki, is a correspondence between the complex representations of a reductive algebraic group $G$ over a local field $F$, and representations of the Langlands group of $F$ into the L-group of $G$.
If we take $G=\mathrm{GL}_2(F)$ where $F$ is a finite extension of $\mathbb{Q}_p$, then by definition, the L-group of $G$ is the semidirect product $^LG^0\times\mathrm{Gal}(\bar{F}/F)$ and the Langlands group of $F$ is the direct product $W_F\times\mathrm{SL}_2(\mathbb{C})$. But when people talk about the local langlands correspondence for $\mathrm{GL}_2$, they usually mean the correspondence between the representation of $W_F$ and the representation of $\mathrm{GL}_2$.
So I'd like to ask why these two statements of local Langlands conjecture for $\mathrm{GL}_2$ are equivalent.
Thanks 🙂
Best Answer
First, the $L$-group is a product in this setting and the map to the second factor is the tautological map from $W_F$ and the trivial map from $\mathrm{SL}_2(\mathbb{C})$ you only have to worry about the first factor. Second, the dual group is just $\mathrm{GL}_2(\mathbb{C})$. Third, a representation of $H \times G$ is just a pair of representations of $H$ and $G$ with commuting image, so we are left with $\rho: W_F \rightarrow \mathrm{GL}_2(\mathbb{C})$ and $\psi: \mathrm{SL}_2(\mathbb{C}) \rightarrow \mathrm{GL}_2(\mathbb{C})$ with commuting image. There are not many possibilities for $\psi$ (more precisely there are exactly two). If $\psi$ it is trivial, then $\rho$ can be anything. But if $\psi$ is non-trivial, then the centralizer of the (irreducible) image consists only if scalars and $\rho$ has to be a character $\chi$.
However, another important correction is that the "usual" correspondence is not with representations $(\rho,V)$ of $W_F$ but Weil-Deligne representations $(\rho,V,N)$. When $\psi$ is non-trivial and so $\rho$ is just a character $\chi$, the associated Weil-Deligne representation $(\rho,V,N)$ is (up to twist depending on your choice of normalizations in local Langlands)
$$V = \chi | \cdot |^{1/2} \oplus \chi | \cdot |^{-1/2}$$
where the nilpotent operator $N$ acts non-trivially. Conversely, any Weil-Deligne representation of dimension $2$ with $N$ non-trivial necessarily has the form $\eta |\cdot| \oplus \eta$ because of how Weil-Deligne representations are defined. So things all match up with the "usual" story.
More generally (for $\mathrm{GL}_n$ instead of $\mathrm{GL}_2$) representations $\psi$ of $\mathrm{SL}_2(\mathbb{C})$ to $\mathrm{GL}_n(\mathbb{C})$ are determined by the conjugacy class of the image of the unipotent element. If the image of $\psi$ is non-trivial, then the image of $\rho(W_F)$ lands inside the centralizer of the image which forces it be be reducible. The corresponding action of $N$ is determined by $\psi$. At one extreme, if $\rho$ is irreducible, then $\rho$ is a character, and the Weil-Deligne representation is a sum of one dimensional representations and $N$ is nilpotent of the largest possible order. If on the other hand $\rho$ is irreducible, then $\psi$ is trivial.