# higher direct image, Shimura varieties of PEL-type and representations

Let $M=M(G,X) = (M_K)_K$ be a Shimura variety of PEL-type associated to datum $(G,X)$. Let $A$ be the universal abelian scheme over $M(G,X)$ and $a: A \rightarrow M$.

Now, using notations from Milne's article from Ann Arbor ("Canonical models..."), take $\rho$ to be a representation of $G^c$. We can associate to it a vector bundle on $M$.

My question is following: is it possible to express $R^q a_* \mathbb{Q} _l$ in terms of the vector bundles associated to representations of $G^c$ as above?

The question comes from the article of Saito "Hilbert modular forms and p-adic Hodge theory", where he says, that it is easy to see using modular interpretation of $M$ ($M$ is not a general PEL-type Shimura variety in the text) that $R^1 a _* \mathbb{Q} _l$ is isomorphic to a vector bundle constructed as above for the very explicit representation (see the beginning of section 6.1). As I don't see it, I would be glad to see an explanation either for this more specific situation or, if the question in general makes sense, I would also be very happy to see a solution for a general Shimura variety of PEL-type.

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The higher push-forwards you're concerned with are $l$-adic local systems. They cannot directly be expressed in terms of vector bundles. What is possible is to look at the associated complex local systems and study them using Hodge theory. It is here that the automorphic vector bundles make an appearance, though there are additional issues when $M$ is not compact. –  Keerthi Madapusi Pera May 19 '11 at 20:36
You might want to look at this question: mathoverflow.net/questions/25354/… –  Keerthi Madapusi Pera May 19 '11 at 20:39
See the article of Saito, which I have mentioned above: he defines by a representation of $G^c$ in (4.8) a local system $\mathcal{F} ' _l$ (if you complexify this, you'll get an automorphic vector bundle) and later on, in the beginning of section 6.1, he claims that $R^1 a_* \mathbb{Q} _l = \mathcal{F} ' _l$. –  Przemyslaw Chojecki May 26 '11 at 13:47
See the article of Saito, which I have mentioned above: he defines by a representation of $G^c$ in (4.8) a local system $\mathcal{F} ' _l$ (if you complexify this, you'll get an automorphic vector bundle) and later on, in the beginning of section 6.1, he claims that $R^1 a_ \mathbb{Q} _l = \mathcal{F} ' _l$ –  Przemyslaw Chojecki May 26 '11 at 13:50