Weil group, Weil-Deligne group scheme and conjectural Langlands group

I was reading a series of article from the Corvallis volume. There are couple of questions which came to my mind:

1. 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.

2. 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.

Regarding (1), from the point of view of Galois representations, the point is that continuous Weil group representations on a complex vector space, by their nature, have finite image on inertia.

On the other hand, while a continuous $\ell$-adic Galois representation of $G_{\mathbb Q_p}$ (with $\ell \neq p$ of course) must have finite image on wild inertia, it can have infinite image on tame inertia. The formalism of Weil--Deligne representations extracts out this possibly infinite image, and encodes it as a nilpotent operator (something that is algebraic, and doesn't refer to the $\ell$-adic topology, and hence has a chance to be independent of $\ell$).

As for (2): Representations of the Weil group are essentially the same thing as representations of $G_{\mathbb Q}$ which, when restricted to some open subgroup, become abelian. Thus (as one example) if $E$ is an elliptic curve over $\mathbb Q$ that is not CM, its $\ell$-adic Tate module cannot be explained by a representation of the Weil group (or any simple modification thereof). Thus neither can the weight 2 modular form to which it corresponds.

In summary: the difference between the global and local situations is that an $\ell$-adic representation of $G_{\mathbb Q_p}$ (or of $G_E$ for any $p$-adic local field) becomes, after a finite base-change to kill off the action of wild inertia, a tamely ramified representation, which can then be described by two matrices, the image of a lift of Frobenius and the image of a generator of tame inertia, satisfying a simple commutation relation.

On the other hand, global Galois representations arising from $\ell$-adic cohomology of varieties over number fields are much more profoundly non-abelian.

Added: Let me also address the question about a product of $W_{F_v}'$. Again, it is simplest to think in terms of Galois representations (which roughly correspond to motives, which, one hopes, roughly correspond to automorphic forms).

So one can reinterpret the question as asking: is giving a representation of $G_F$ (for a number field $F$) the same as giving representations of each $G_{F_v}$ (as $v$ ranges over the places of $F$). Certainly, by Cebotarev, the restriction of the global representation to the local Galois groups will determine it; but it will overdetermine it; so giving a collection of local representations, it is unlikely that they will combine into a global one. ($G_F$ is very far from being the free product of the $G_{F_v}$, as Cebotarev shows.)

To say something on the automorphic side, imagine writing down a random degree 2 Euler product. You can match this with a formal $q$-expansion, which will be a Hecke eigenform, by taking Mellin transforms, and with a representation of $GL_2(\mathbb A_F)$, by writing down a corresponding tensor product of unramified representations of the various $G_{F_v}$. But what chance is there that this object is an automorphic representation? What chance is there that your random formal Hecke eigenform is actually a modular form? What chance is there that your random Euler product is actually an automorphic $L$-function? Basically none.

You have left out some vital global glue, the same glue which describes the interrelations of all the $G_{F_v}$ inside $G_F$. Teasing out the nature of this glue is at the heart of proving the conjectured relationship between automorphic forms and motives; its mysterious nature is what makes the theories of automorphic forms, and of Galois representations, so challenging.

The answer to your first question would be a Steinberg representation (i.e. under suitable normalizations, the infinite-dimensional subquotient of the induction of $(\chi|\cdot|^{-1/2},\chi|\cdot|^{1/2})$). Kudla's article in Motives II is a nice place to see this. I don't have an answer for number two.

• I see, since it corresponds to a nontrivial nilpotent operator, it can not come from a representation of Weil group. Or is there some other argument? Thanks – Dipramit Majumdar Feb 28 '10 at 5:07

(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.

• Actually the Steinberg representation $\sigma$ is ramified of conductor $1$, and its $L$-function equals $(1 - \chi(\varpi)q^{-s-1/2})^{-1}$. In general, let $\pi$ be a local irreducible generic representation of $\mathrm{GL}_n$. Then $L(s,\pi)^{-1}$ is a polynomial of $q^{-s}$ of degree at most $n$, and the degree equals $n$ if and only if $\pi$ is unramified. See the Proposition on page 203 of Jacquet-PiatetskiShapiro-Shalika: Conducteur des répresentations du groupe linéaire (Math. Ann. 256 (1981), 199-214). – GH from MO Jul 31 '13 at 17:12
• Fixed. Thank you for pointing out my error. Apparently my understanding of the Artin conductor is limited, looks like it's time to go back and read Tate's Corvallis article again. As a side question, my construction depends on the ideas of Bernstein-Zelevinski. Does anyone know of a construction of a "counterexample" that does not rely on these ideas? – John Binder Jul 31 '13 at 18:31