On a claim of Deligne about representations of Weil-Deligne groups

Question

In Deligne's article "Les constantes des equations fonctionelles des fonctions L", we find the following claim:

Proposition 8.9 (ibid.): Suppose $(V, \rho, N )$ and $(V', \rho', N')$ are two Frobenius semisimple, $\ell$-adic and $\ell'$-adic representations of the Weil-Deligne group $' W_K $ of a (non-archimedean ) local field $ K$, respectively. Let $\mathbf{M}_{\bullet}$ and $\mathbf{M'}_{\bullet}$ be the (monodromy) filtrations induced by $N $ and $N'$ on $ V$ and $V'$, respectively. Then $(V, \rho, N )$ and $(V', \rho', N')$ are compatible (in the sense of ibid. 8.7) if and only if the characters, of the representations induced by $\rho$ and $\rho '$ on the graded parts of $\mathbf{M}_{\bullet}$ and $\mathbf{M'}_{\bullet}$ coincide and has values in $\mathbb{Q}$.

Deligne states this as an exercise. Does anyone know a rigorous proof or any reference?

Of course it is clear that if the characters of the representations (as in the proposition) induced by $\rho $ and $ \rho'$ on the graded parts of the filtrations $\mathbf{M}_{\bullet}$ and $\mathbf{M'}_{\bullet}$, coincide and has values in $\mathbb{Q}$, then the character of $\rho $ and $ \rho'$ are same and hence due to semi-simplicity, $\rho $ and $\rho'$ are isomorphic (after base extension to a common algebraic closure $\Omega \supset \mathbb{Q}_{\ell}, \mathbb{Q}_{\ell '} $) as representations of the Weil group $ W_K $. But what is not clear to me is how to find an isomorphism $ f $ between $ \rho$ and $ \rho' $ ( after base extension $\Omega $), such that

$$ f \circ N = N' \circ f \quad \text{(after base extension to $\Omega $) } \;? $$

Answer 1

The answer was completed with help from Prof. Pierre Deligne.

First we have a group $G=F^{\mathbb Z}\ltimes I$, a representation $(V,\rho)$ of $G$ over a field of characteristic zero such that $I$ acts through a finite quotient, and that $\rho(F)$ is semi-simple. We also have a nilpotent endomorphism $N$, connecting with $I$ and such that $\rho(F)$ conjugates $N$ into $q^{-1}N$.

We have to show that $(V,\rho,N)$ is determined up to isomorphism by the isomorphism clone of the representations $\mathfrak S_j^{\bf M}(V)$ of $G$. The key of the proof is in the following picture (sorry, it’s hard for me to draw it in $\LaTeX$): the $\bullet$ stands for representations of $G$; $V$ is the sum of these representations; $N$ is given by vertical arrows; $\mathbf{M}_j$ is what is up to the line $j$; given how $F$ acts on $N$, a $\bullet$ under another is a Tate twist of it.

Let us first forget about $\rho$. Such a picture for $N$ alone follows from the Jordan normal form, or from the Jacobson-Morozov theorem: columns are isotypical representations of $\mathrm{SL}_2$.

Now look at the choice required to put $N$ into this form. For each $j\ge 0$, we should look at the primitive part $P_j\subset \mathfrak S_j^\mathbf M(V)$ $$P_j=\ker(N^{j+1}:\mathfrak S_j\to \mathfrak S_{j-2})$$ and lift it in $\ker(N^{j+1}:V\to V)\subseteq {\bf M}_j$ $$\require{AMScd}\begin{CD} \\ @.\tilde{P}_j@>>\sim>P_j\\ @VV\subset V @VV\subset V\\ @.\mathbf{M}_j@>>>\mathbf{M}_{j+1} \end{CD}$$ where $N^{j+1}\tilde{P}_{j}=0$. Then we get the picture, the $\bullet$ being $N^a\tilde{P}_j$. Then we put $\rho$ back in. We have:

• $(V,\rho)$ is semi-simple;
• $\ker(N^{j+1}:\mathbf{M}_j\to \mathbf{M}_{-j-2})$ is a representation of $G$, mapping to the representation $\mathfrak{S}_j^{\mathbf{M}}(V)$ with image $P_j$. By semi-simplicity, we can lift $P_j$ on $\tilde{P}_j\overset{\sim}{\longrightarrow} P_j$. Taking again $N^a\tilde{P}_j$, we get the picture with the $\bullet$ being now representation over $G$.

The isomorphism clone of $(V,\rho,N)$ is hence determined by that of the $P_j$, and it remains to use that $$P_j\oplus \mathfrak S_{j+2}^\mathbf M(V)\xrightarrow[\rm bijective,\ N]{\sim} \mathfrak S_{j}^\mathbf M(V)$$ so that the isomorphism clone of $P_j$ is given by $$\mathfrak S_j — \mathfrak S_{j+2}$$ which makes sense by semi-simplicity.