Let $G$ be a connected reductive group over a number field $K$, $P$ be a parabolic subgroup of $G$ defined over $K$, $X=G/P$ be the generalized flag variety which is a smooth projective variety over $K$ and $p$ be a prime number. For a positive integer $i>0$, consider the etale cohomology $V=H^i(X_{K^{alg}},\mathbb Q_p)$ as a Galois representation of $G_K$.

How to compute such Galois representation? This may be done somewhere but I can't find a reference. Firstly, the dimension may be computed by using Betti numbers and some combination datas from the Lie algebra (the odd dimension shall vanish).

Secondly, is the representation semi-simple? For the projective space it's obviously true as the dimension is no bigger than $1$. If that's true, then must the direct summand be some $\mathbb Q_p(-i/2)$? Some density theorem may reduce this to the finite field case.

In a short word, how to completely decide the Galois representation? As the flag variety has a stratification by affine spaces, this seems reachable.