Given a reductive group $G/\mathbf Q$ (+ additional data), and a compact open subgroup $K\subset G(\mathbf A^\infty)$, there is a standard construction that produces a Shimura variety $S$ and if we choose a rational representation $\xi : G\to Aut(V)$, we obtain a certain $l$-adic local system $\mathcal L$ on $S$. Moreover, all of this can be done over some fixed number field $E$.
Question 1: If $S$ is proper, is it true that the $Gal(\overline E/E)\times H(G,K)$ action on $H^*(S\otimes\overline E, L)$ is semisimple? (where $H(G,K)$ is the corresponding Hecke algebra).
Question 2: What happens if $S$ is not proper? Is the action semi-simple at least if we restrict to parabolic cohomology (the image of compactly supported cohomology inside usual cohomology)?
References for places where this is discussed would also be great.