Let $G$ be a $\mathbb{Q}$-subgroup of $\mathrm{GSp}\_{2g}$, reductive and defines a Shimura subdatum of $(\mathrm{GSp}\_{2g},\mathfrak{H}\_g)$. Let $V$ be the natural representation of $\mathrm{GSp}\_{2g}$ (so $V$ is a $\mathbb{Q}$-vector space of dimension $2g$). Assume $V$ is irreducible as a $G$-module, then is $V\_{\mathbb{R}}$ also irreducible as a $G\_{\mathbb{R}}$-module?


In other words, the family of abelian varieties parametrized by the Shimura subvariety defined by $G$ cannot be decomposed into a product, but I want to know whether the family of the tangent spaces of these abelian varieties can be written in a product.



Tangent spaces of an indecomposable family of abelian varieties (parametrized by a Hodge type Shimura variety)
No. Here is a counter-example (I add details to my comment).


Let $F$ be a totally real number field. 
Consider the group $G\_0=R\_{F/\mathbb{Q}}\mathrm{SL}\_{2,F}$, then $G\_0$ naturally embeds into $\mathrm{Sp}\_{2g}$, where $g=[F:\mathbb{Q}]$.
Set $G=\mathrm{G}\_{m,\mathbb{Q}}\cdot G\_0$, then $G$ naturally embeds into 
$\mathrm{GSp}\_{2g}=\mathrm{G}\_{m,\mathbb{Q}}\cdot \mathrm{Sp}\_{2g}$.
The pair $(G,X)$ with suitable $X$ is a Shimura subdatum of $(\mathrm{GSp}\_{2g},\mathfrak{H}\_g)$, see Deligne's papers.


On the other hand, the representation of $G$ in $V=R\_{F/\mathbb{Q}} F^2$ is $\mathbb{Q}$-irreducible, while the representation of $G\_{\mathbb{R}}$ in $V\_{\mathbb{R}}$ is a direct sum of $g$ two-dimensional representations.


