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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

flag	asked Mar 8 2011 at 16:31 jacob 355●3●6

Keerthi Madapusi Pera · Accepted Answer · 2011-03-08 21:17:47Z UTC

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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

answered Mar 8 2011 at 21:17

Keerthi Madapusi Pera
1,736●1●5●10

	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 2011 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 2011 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 2011 at 11:37
	Ah, yes, they must be of pure $D$-type. Thank you for the correction. – Keerthi Madapusi Pera Mar 9 2011 at 15:54

Are all Shimura Varieties Special Subvarieties of the Siegel modular Variety?

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