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Let $S _K = S_K(G,X)$ be a Shimura variety of dimension $n$. Let $\xi$ be a (finite-dimensional) representation of $G$, which gives rise (by a construction of Harris) to an automorphic bundle $V(\xi)$ on $S_K$.

What is known about the vanishing of $H^i(S_K, V(\xi))$?

What I would like to know ideally, is whether we have (for example, for compact Shimura varieties of PEL-type) $H^i(S_K, V(\xi)) = 0$ for each $i \not = n$? (possibly, after a localisation at some "non-Eisenstein" ideal). Could we expect it in general?

I am aware of results of Lan-Suh and Mokrane-Tilouine (who proved this result in some special cases under certain conditions on $\xi$), though I still have not digested it properly. Any references and comments are welcome!

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