Semisimplicity of étale cohomology representations

This semi-simplicity is a part of what is called the Tate conjecture. It is generally believed to be true, but little is known about it outside the case of $H^1$, in either the finite field or global field case. Searching on mathscinet for "Tate conjecture" (or googling) should turn up the relevant literature.

In https://arxiv.org/pdf/1709.04489.pdf, Moonen proves that for finitely generated fields of characteristic $0$, the Tate conjecture (surjectivity of the cycle class map $\mathrm{CH}^r(X) \otimes \mathbf{Q}_\ell \to \mathrm{H}^{2r}(\bar{X},\mathbf{Q}_\ell)^{G_K}$) implies the semisimplicity conjecture.

(For the Tate conjecture, see e.g. http://www.math.columbia.edu/~chaoli/docs/TateConjecture.html or Tate's article in the Motives volume.)