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



Are all Shimura Varieties Special Subvarieties of the Siegel modular Variety?
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](http://www.ams.org/mathscinet-getitem?mr=0546620), following Satake [here](http://www.jstor.org/stable/2373012). See also Proposition 1.21 in Milne's article [here](http://www.jmilne.org/math/articles/1994bP.pdf) 


