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.



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


