Semisimple Weil-Deligne representations I've just realized that I don't understand something important and basic about the Weil-Deligne group and its representations.  (I'm not very surprised by this).
Following Deligne's article, Section 8 of "Les Constantes des Equations Fonctionnelles Des Functions L" from the 1972 Antwerp volume, one is led to consider representations of the Weil-Deligne group in the following algebraic sense:  for any local nonarchimedean field, there is a group scheme $W'$, defined over $Q$, which is a semidirect product of the Weil group scheme $W$ and the additive group scheme $G_a$.  This is a non-affine group scheme, and the Weil subgroup scheme $W$ is obtained as a countable disjoint union of affine subschemes (the cosets of inertia).
So, as is now standard, we consider algebraic representations of the group scheme $W'$, over various fields $E$ of characteristic zero, as such representations provide a unified framework (thanks to results of Grothendieck, Deligne, Serre) for $\lambda$-adic representations that arise from arithmetic.
A crucial piece of this is to restrict attention (or semisimplify) to the semisimple representations of the Weil-Deligne group.  And presumably, such semisimple representations form a Tannakian category (over any base field of characteristic zero).
And so to my question... what is the algebraic group associated to this Tannakian category?  Or am I just confused?  And how does the (non-affine) Weil-Deligne group scheme relate to this (affine) algebraic group obtained by restricting attention to these semisimple representations?  Does this involve one of these awfully large group schemes like $Spec(E[E^\times])$, where $E$ is a characteristic zero field (something like the semisimple algebraic hull of the discrete group $Z$)? Any references?
 A: Is this really a sensible question to ask, I wonder?
Here's a guess as to what the answer might look like. The Weil-Deligne group comes in three pieces. First there's inertia. Then there's a copy of $\mathbf{Z}$. And finally there's your $N$. Now the $N$ piece works fine: that will contribute something like an affine line (considered as additive group) to the situation. But the other two pieces are surely monstrous. The inertia action is via a finite group, by definition, and so you'll take something like the affine group schemes corresponding to these finite groups and then take some monstrous projective limit. And the $\mathbf{Z}$ part, because it's semisimple by definition, is basically a grading of your vector space by non-zero elements of the ground field, so it too will be monstrous: you seem to be already aware of the monsters that occur when you try and write these things down explicitly: a $\mathbf{G}_m$ gives a $\mathbf{Z}$-grading, some funny projective limit of such things gives a $\mathbf{Q}$-grading, now raise this to some uncountable cardinal $I$ to get a $\mathbf{Q}^I$-grading, because as an abelian group $\mathbf{C}^\times$ is quite close to $\mathbf{Q}^I$ for some uncountable $I$, but then there's some torsion so add some $\mu_n$'s for the $n$-torsion etc etc and you get the sort of group you mention in the question.
And finally glue them all together carefully. 
