Consider the Shimura datum $(GSp\_{2g},\mathcal{H}\_g)$. Let $G$ be a reductive $\mathbb{Q}$-subgroup of $Sp\_{2g}$. I want to know under what condition there exists a point $x\in\mathcal{H}\_g$ such that $h\_x\colon\mathbb{S}\rightarrow GSp\_{2g,\mathbb{R}}$ factors through $Z(GSp\_{2g})G$.


There are some easy necessary conditions: $G$ shouldn't be of mixed type $D\_n$, *$G(\mathbb{R})$ should be non compact* and $Z(G)(\mathbb{R})$ should be compact. **My first question is whether these conditions are sufficient.**


My second question is: can a $\mathbb{Q}$-reductive subgroup of $Sp\_{2g}$ be of mixed type $D\_n$?



Subgroups of $Sp_{2g}$ giving rise to Shimura data
First question: No, these conditions are not sufficient. For details see: Deligne, Pierre, Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques. (French) Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, pp. 247–289, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979.


Second question: Yes, any linear algebraic $\mathbb Q$-group can be embedded into $GL\_n$ for some $n$, and $GL\_n$ can be embedded into $Sp\_{2n}$.


