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