Given a reductive group $G/\mathbf Q$, 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.

Thanks

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.

Thanks

Given a reductive group $G/\mathbf Q$, 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?

Thanks

Given a reductive group $G/\mathbf Q$, 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.

Thanks