# 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?

• Why is a tensor product of semisimple representations semisimple? – Bruce Westbury Mar 26 '11 at 8:07
• For this semisimplicity condition to be satisfied, it is necessary and sufficient for a Frobenius element $\phi$ to go to a semisimple element of $GL_n$. The "Kronecker" map sends a pair $(g,h)$ of semisimple elements of $GL_m$ and $GL_n$ to a semisimple element of $GL_{mn}$ -- easy to prove over alg. closed field by conjugacy of tori, then use descent. – Marty Mar 26 '11 at 15:21
• @Bruce: here is a striking theorem of Chevalley. Let $G$ be any group and let $E$ be a field of char $0$ and let $V$ and $W$ be two finte dimensional $E$-linear reps of $G$. If $V$ and $W$ are semisimple, then so is $V \otimes_E W$. – Laurent Berger Apr 17 '11 at 8:36

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.
• PS I guess here is what is confusing me: you are saying "I know that the category of semisimple representations of $\mathbf{Z}$ is Tannakian and the corresponding affine group is horrible. Now consider a group $W$ which surjects onto $\mathbf{Z}$ and connsider the corresponding affine group. Will there be an explicit relation between $W$ and the affine group, other than "semisimple algebraic hull", even though there already seems to be no such relation for $\mathbf{Z}$?". Perhaps I've misunderstood what you're asking though? – Kevin Buzzard Mar 26 '11 at 16:44
• Maybe it's not a sensible question, and your response confirms my suspicions. I had some recollection of seeing a group scheme like $Spec(C[C^\times])$, but I don't recall a reference. Maybe you know where to read about such things? Your comments about the "Z part" being "a grading of your vector space by non-zero elements of the ground field" helps me to understand the awfulness involved. So I can work it out from here, even if I can't find a reference. – Marty Mar 26 '11 at 18:30
• Marty: Yeah, the Tannakian category of representations of $GL(1)$ is the category of $\mathbf{Z}$-graded vector spaces so there's a nice example. But the category of $\mathbf{Q}$-graded vector spaces is already terrifying: you take some projective limit of $GL(1)$s under the maps $z\mapsto z^{n!}$ or whatever and already you have something quite icky. My guess is that knowing you have an affine group scheme can be helpful when doing theoretical stuff, but my experience of the Tannakian categories showing up in Langlands' stuff is that the group scheme is quite humungous and weird in general. – Kevin Buzzard Mar 26 '11 at 20:07
• PS perhaps the category of $A$-graded complex vector spaces ($A$ any ab group) is the reps of the group $Spec(\mathbf{C}[A])$ as you say? The only thing I ever read about Tannakian categories was the stuff by Deligne and Milne. That was a pretty clear paper, I thought. My memory is that the $GL(1)$ example and the proj limit were already mentioned in that. I don't know much about this stuff really. – Kevin Buzzard Mar 26 '11 at 20:10