A question related to the semisimplification of a Weil-Deligne representation

Question

I have been trying to find the answer to this question, I think it must not be hard but I don't get it.

I have a Weil-Deligne representation ($\rho,N$) of the Weil group $W$ of $Q_p$, that is $\rho$ is a representation in $V$ of $W$ in and $N$ is a (nilpotent) endomorphism of $V$ verifying

$$ \rho(w)N=\rvert w\rvert N\rho(w). $$ Taking a frobenius $\Phi$ of $W$ and the decomposition $\rho(\Phi)=su$ with semisimple $s$ and unipotent $u$. It can be shown that $N$ and $u$ commute. This part is not clear to me. Does someone has any idea of why this is true?

Thank you in advance.

Answer 1

The assumption says ${}^{su} N = |w| \, N$. That is, $N$ is an eigenvector (with eigenvalue $|w|$) for the action of $\rho(w) = su$ by conjugation on $End(V)$. Since $su$ is the Jordan decomposition, this implies that $N$ is fixed by $u$, as desired.