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Given a Shimura variety $S$, is it possible to imbed $S$ as a special Subvariety of the Siegel modular variety $A_{g,N}$, for some $g$ and level $N$? I expect that the answer is yes, essentially since every semisimple group over $\mathbb{Q}$ should imbed into $GL_n$ via its adjoint representation, and $GL_n$ imbeds into $SP_{2n}$. However, I'm a bit worried about the business regarding weights.

Thank you, Jacob

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up vote 4 down vote accepted

The answer is no, in general. The problem is to find an embedding so that the minuscule character corresponding to the Shimura datum for $S$ induces the minuscule character of $GSp_{2n}$ corresponding to a decomposition into Lagrangians.

In the affirmative direction, for most classical, simply connected groups (and only for classical groups, i.e. of types $A$,$B$,$C$ and $D$), the answer is yes; some subtleties crop up for $Spin^*(2n)$ (this is the so called $D^{\mathbb{H}}$ case), for which only the quotient by an order 2 central sub-group admits a symplectic embedding (of Shimura data).

This is all beautifully laid out in Deligne's article 'Varietes de Shimura...' here, following Satake here. See also Proposition 1.21 in Milne's article here

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Thanks! Just so I understand, for the classical groups, the answer is yes independantly of the $\mathbb{Q}$-structure? So for instance, for any Shimura variety arising as a quotient of the Siegel upper half plane? – jacob Mar 8 '11 at 22:41
Yes, the question doesn't depend on the $\mathbb{Q}$-structure; see 2.3.10 in Deligne's article. You have to be careful to distinguish between a Shimura variety and its (geometrically) connected components. In general, only the latter can be quotients of the upper half plane. – Keerthi Madapusi Pera Mar 9 '11 at 5:28
It is not quite correct that the answer does not depend on the $\mathbb{Q}$-structure. For example, let $G$ be a $\mathbb{Q}$-group such that the decomposition of $G_{\mathbb{R}}$ into a product of simple $\mathbb{R}$-groups contains both $\mathrm{SO}(8,2)$ and $\mathrm{SO}^*(10)$ (and only these $\mathbb{R}$-groups). If $G$ is a product of $\mathbb{Q}$-abolutely-simple groups, then a Shimura datum $(G,X)$ admits a symplectic embedding. However, if $G$ is $\mathbb{Q}$-simple, then $(G,X)$ has no symplectic embeddings. – Mikhail Borovoi Mar 9 '11 at 11:37
Ah, yes, they must be of pure $D$-type. Thank you for the correction. – Keerthi Madapusi Pera Mar 9 '11 at 15:54

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