TITLE: Are all Shimura Varieties Special Subvarieties of the Siegel modular Variety?
QUESTION [5 upvotes]: 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

REPLY [4 votes]: 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