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.

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