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.