Let $K$ be a number field and $G=Gal(\overline{K}/K)$ the absolute Galois group of $K$. Let $\ell$ be a prime number.


Let $A/K$ be an abelian variety. Then the representation of $G$ on $V\_\ell(A)$ is semisimple. This is the famous theorem of Faltings (Invent. Math. 73).


Now let $X/K$ be a smooth projective variety and $0\le q\le 2\dim(X)$, and define $\overline{X}=X\_{\overline{K}}$.


**Question.** Is it known that the representation of $G$ on $H^q(\overline{X}, \mathbb{Q}\_\ell)$is semisimple?


**Remark.** The answer is yes for $q=1$, because $H^1(\overline{X}, Q\_\ell)$ is dual to
$V\_\ell(A)$ where $A$ is the Albanese variety of $X$.


I would also be interested in the case where the number field $K$ is replaced by a global function field (say), and $\ell$ is assumed to be coprime to the characteristic. 



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. 


