While reading Lemma 3 of this paper, I encountered the following statement:
Take a sufficiently large finite Galois extension $F/K$ such that $\rho/F$ is unramified. Then $\rho(\operatorname{Frob}_F )$ is central in $\rho(W_K)$, so the eigenspaces of $\operatorname{Frob}_F$ are $W_K$-subrepresentations.
I know that:
- $K$ is a local field, and so is $F$ I think,
- $W_K$ is the Weil group of $K$,
- $\rho$ is a Frobenious-semisimple Weil presentation over $K$
- $Frob_F$ is an arithmetic Frobenius element.
But what exactly does "central" mean in this context? I have never encountered this term before.
It means that the image of $\mathrm{Frob}_F$ is in the centre of the image of $\rho$.
If $\rho:W_K\to \mathrm{GL}_n(\mathbb C)$ is a Weil representation, then the image of inertia is finite, so there is a finite extension $F/K$ such that $\rho|_F$ is unramified.
So $\rho|_F$ factors through $W_F/I_F =\langle\mathrm{Frob}_F\rangle$. A priori, $\rho(\mathrm{Frob}_F)$ could be something like $\begin{pmatrix} 1&1\\0&1\end{pmatrix}$. However, the fact that $\rho$ is Frobenius semisimple, combined with Schur's lemma, means that $\rho(\mathrm{Frob}_F)$ must act by a (one-dimensional) character. In particular, it lies in the centre of the image of $\rho$.