Central isogeny, Shimura varieties and exceptional cases

Question

For a simple complex Lie algebra $\mathfrak g$, its weight lattice is not equal to the root lattice (i.e. the center of its simply connected form is a non-trivial finite group) iff $\mathfrak g$ is of type $A, B, C, D, E_6, E_7$. And we know Shimura varieties exist exactly in type $A, B, C, D, E_6, E_7$, which is quite mysterious. Note that $G_2, F_4, E_8$ are simply connected.

Here are my questions:

1. given a type $A, B, C, D, E_6, E_7$, what are exactly these Shimura data $(G, X)$ with the given type (i.e. the derived subgroup of $G$ has Lie algebra with given type)? Here $G$ is a reductive group over $\mathbb Q$, and we only focus on the structure of $G_\mathbb R$ as a real reductive group. Of course, we could always take products with Shimura datum for a torus. In particular, for $E_6$-Shimura varieties, could we choose the simply connected $E_6$ or adjoint type $E_6$ (plus some torus)?

2. Among these choices, which of them has the smallest reflex field? What is the reflex field (e.g. for $E_6$ and $E_7$)?

3. Let $f: G_1 \to G_2$ be a central isogeny of reductive groups over $\mathbb Q$. When $(G_2, X_2)$ is a Shimura datum, when could we extend $f$ to a map between Shimura datum $(G_1, X_1) \to (G_2, X_2)$?

Motivation: there is a construction of adjoint Shimura datum from any Shimura datum. However, $G(\mathbb R) \to G^{ad}(\mathbb R)$ may not be surjective. We often talk modular curves for $G=GL_2$ (rather than $G=PGL_2$ or $G=SL_2$), Siegel modular varieties for $G=GSp_{2n}$ (rather than $G=Sp_{2n}$, or $G=PSp_{2n}$), unitary Shimura varieties ($G=GU_n$, $G=U_n$, $G=SU_n$). It seems that central isogenies will change our understanding of Shimura varieties in a big way (e.g. we may lose naive moduli descriptions).

A full classification of Shimura datum seems hard to find in the literature (although for conjectures like Andre-Oort we can always pass to adjoint Shimura datum). I only know the recent paper of Ben Moonen https://arxiv.org/abs/2405.20673v1 which gives a precise classification of all $1$-dimensional Shimura datum inside Siegel Shimura datum $(GSp_{2n}, \mathbb H_n^{\pm})$.

Answer 1

You can find answers to your questions in: Deligne, Pierre: Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques. 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, American Mathematical Society, Providence, RI, 1979. Here I give some hints.

1. See Deligne, Table 1.3.9 on page 261. Yes, for $E_6$ we can choose the adjoint $E_6$. Yes, for $E_6$ we can choose a reductive ${\Bbb Q}$-roup $G$ that fits into a nonsplit short exact sequence $$ 1\to G^{\rm der} \to G\to T\to 1$$ where the derived group $G^{\rm der}$ is an absolutely almost simple, simply connected group of Hermitian type of type $E_6$, and $T$ is a $\Bbb Q$-torus. I think that we cannot take $G=G^{\rm der}\times T$ when $G^{\rm der}$ is not adjoint.

2. See Deligne, 2.2.1 on page 268. The smallest reflex field is for the adjoint group. If $G$ is absolutely simple adjoint over $\Bbb Q$ of type $E_7$, then $E(G,X)=\Bbb Q$. For $E_6$ I think we will have an imaginary quadratic field. See Deligne, Table 1.3.9.

3. No. Take $G_2={\rm PGL}_2$, $G_1={\rm SL}_2$.