TITLE: Weil group, Weil-Deligne group scheme and conjectural Langlands group
QUESTION [19 upvotes]: I was reading a series of article from the Corvallis volume. There are couple of questions which came to my mind:

Why do we need to consider representation of Weil-Deligne group? That is what is an example of irreducible admissible representation of $ Gl(n,F)$ which does not correspond to a representation of $W_F$ of dimension $n$ ? An example for $ n=2 $ will be of great help.
In the setting of global Langlands conjecture, why extension of $W_F$ by $G_a$ or products of $W'_{F_v}$ does not work? 

Thank you.

REPLY [3 votes]: (I'm putting an "answer" to clarify Rob's question, and answer Dipramit's question in the comments, because I don't yet have the reputation to comment).
Let's first recall that the L-function of the Steinberg representation $\sigma = \sigma(\chi|\cdot|^{-1/2},\, \chi|\cdot|^{1/2})$ (for $\chi$ an unramified character) is $(1 - \chi(\varpi)q^{-s-1/2})^{-1}$, where $\varpi$ is a uniformizer (Bump shows this in detail in his book).  In particular, its reciprocal is a degree-one polynomial in $q^{-s}$.
By the ideas of Bernstein-Zelevinski (described in Kudla' article in "Motives"), $\sigma$ corresponds to the Weil-Deligne representation $\rho' = (\rho,\, V,\, N)$, where $\rho = \chi |\cdot|^{-1/2} \oplus \chi |\cdot|^{1/2}$, and the operator $N$ takes the first summand to the second, and the second summand to $0$.
If we only look at $\rho$, then we see that $V^I = V$ and therefore the $L$-function of $\rho$ would be the reciprocal of a degree-two polynomial.  Thus the monodromy operator becomes necessary for match of $L$-function: we see that $V^I_N \cong \chi|\cdot|^{1/2}$, which has the desired $L$-function.
This is a good specific example if you like thinking about match of $L$-functions.  More generally, you need to consider Weil-Deligne representations because if $\pi_1$ and $\pi_2$ have the same cuspidal support and correspond to $\rho_1' = (\rho_1,\, N_1,\, V_1)$ and $\rho_2' = (\rho_2,\, N_2,\, V_2)$, then $\rho_1\cong \rho_2$ as Weil representations; this follows from the ideas of Bernstein-Zelevinski.