# Tangent spaces of an indecomposable family of abelian varieties (parametrized by a Hodge type Shimura variety)

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.

• No. Take $G=\mathrm{G}_{m,\mathbb{Q}}\cdot R_{F/\mathbb{Q}}\mathrm{SL}_{2,F}$, where $F$ is a totally real extension of $\mathbb{Q}$. Here $g=[F:\mathbb{Q}]$. – Mikhail Borovoi Apr 18 '16 at 6:26

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.