On the semisimplicity conjecture and Galois representations

The semisimplicity conjecture says that for any smooth projective scheme $X_0$ over a finite field $\mathbf{F}_q$, the Frobenius correspondence acts semisimply on $H^i(X\otimes_{\mathbf{ F}_q} \mathbf{ F}, \overline{\mathbf{ Q}}_l)$, where $\mathbf{ F}$ is an algebraic closure of $\mathbf{ F}_q$. Based on the works of Deligne and Laumon, we reduce this conjecture to a problem about the Galois representations of function fields. This reduction was also achieved by Laumon a few years ago (unpublished).