Suppose $X$ is a smooth projective variety defined over $\mathbb{Q}$, and the pure Hodge structure on $H^{2n}(X)$ is of a very simple form,
\begin{equation}
\mathbb{Q}(-n)^{b^{2n}}
\end{equation}
where $b^{2n}$ is just $\text{dim}\,H^{2n}(X)$. Under what conditions is the etale cohomology $H^{2n}\_{et}(X,\mathbb{Q}\_{\ell})$ also isomorphic to 
\begin{equation}
\mathbb{Q}\_{\ell}(-n)^{b^{2n}}
\end{equation}
as representations of $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$?


This is however not always true, consider a smooth quadratic surface $S$ of $\mathbb{P}^3\_{\mathbb{Q}}$ defined by a quadratic polynomials with determinant say 7. Then the Hodge structure on $H^2(S)\simeq H^2 (\mathbb{P}^1 \times \mathbb{P}^1)$ is $\mathbb{Q}(-1)^{2}$, while $H^{2}\_{et}(S,\mathbb{Q}\_{\ell})$ is 
\begin{equation}
\mathbb{Q}\_{\ell}(-1)^{2} \otimes \chi
\end{equation}
where $\chi$ is a Dirichlet character.



When is the etale cohomology of a variety simple if its Hodge structure is simple?
Assuming the Hodge conjecture, it's true if and only if the relevant algebraic cycles are defined over $\mathbb{Q}$. 
Conjecturally, the cohomologies are different realizations of the same motive. The condition on the Hodge cohomology tells you the motive is a Tate twist of an Artin motive, and you want the Artin motive to be trivial.


