It is conjectured that the action of the Frobenius acting on the etale cohomology of an algebraic variety over a finite field is semisimple. A first approximation of my question is :

What is the importance of this statement ?

It is only an approximation because this question is rather unprecise. More precisely, I would like to know two different things :

1) Is there some conjectural statements which are implied by the semisimplicity conjecture ?

More generally :

2)What the semisimplicity property means ?

I ask for an arithmetic or geometric interpretation of the semisimplicity of the Frobenius. For example, the eigenvalues of the Frobenius are important because they control the number of points of the variety over finite fields. I have the impress that often people only care of eigenvalues of the Frobenius (often there are remarks of the form : "if the representation is not semisimple, take the semisimplification"). Is there some situation where the conjectural semisimplicity would be an important thing ?

Remark : I don't want an answer like "it is important because it is implied by the standard conjectures and so if it would be true it would be a confirmation of a part of a general philosophy ..." I would like some "concrete" example of the use of the semisimplicity property (maybe in the cases where it is actually known : abelian varieties ...)