It is clear that for a given connected reductive $\mathbb{Q}$-group $G$, there are at most finitely many Shimura data of the form $(G,X)$ up to isomorphism. But in what case is it unique?

Moreover, in a given Shimura datum $(G,X)$, say we have a subdatum $(H,X_H)$, is it possible to find other subdatum of the form $(H,X'_H)$ with $X'_H\neq X_H$ as subsets of $X$? Of course there are a priori only finitely many choices for such $X_H'$, but I doubt if one is essentially reduced to the trivial case.

thanks a lot!

A pure Shimura datum is of the form $(G,X)$ with $G$ a connected reductive $\mathbb{Q}$-group, and $X$ a homogeneous space under $G(\mathbb{R})$, subject to Deligne's conditions in terms of Hodge types, Cartan involutions, etc. cf.Deligne, "Varietes de Shimura: interpretation modulaire, et techniques de constructions de modeles canoniques"

One may ask to a given connected reductive $\mathbb{Q}$-group, how many pure Shimura data could one obtain of the form $(G,X)$. As remarked by M.Borovoi, if $G$ is a compact $\mathbb{Q}$-torus, then any homomorphism $h:\mathbb{C}^\times/\mathbb{R}^\times\rightarrow G_{\mathbb{R}}$ defines a pure Shimura datum $(G,\{h\})$. Thus for a finiteness answer, one should restrict to the case where $G$ is semi-simple.

For simplicity one even assume that $G$ is of adjoint type. Then as is pointed out in Deligne's article, the existence of $X$ is characterized by the special nodes in the Dynkin diagram of $G_\mathbb{C}$ (plus certain condition so that the node gives rise to an $\mathbb{R}$-homomorphism $\mathbb{S}\rightarrow G_\mathbb{R}$. For the adjoint $G$ chosen, there are at most finitely many special nodes (possibly empty in certain cases). And thus the finiteness is clear.

My first question is: for a given adjoint $\mathbb{Q}$-group $G$ and some pure shimura datum $(G,X)$, how many Shimura data can one obtain to be of the form $(G,X')$ (with the same $G$)$ up to isomorphism? When is it unique? (added: M.Borovoi has answered this in detail, see below).

Secondly, what about Shimura subdatum in a fixed $(G,X)$ can one find to share the common underlying $\mathbb{Q}$-group? By a Shimura subdatum is meant a pair $(G_1,X_1)$ where $G_1$ is a $\mathbb{Q}$-subgroup, $X_1$ a $G_1(\mathbb{R})$-orbit in $X$ such that $(G_1,X_1)$ is a Shimura datum itself. it is also easy to check that to obtain such a aubdatum, it suffices to (1) find a point $x$ in $X$ such that the corresponding homomorphism $h:\mathbb{S}\rightarrow G_\mathbb{R}$ has image in $G_{1, \mathbb{R}}$.

And my second question is: if a connected reductive $\mathbb{Q}$-subgroup $G'$ of $G$ is given, how many subdatum of $(G,X)$ can one find to be of the form $(G',X')$? If there are such subdata, are they unique up to isomorphism (or isomorphism induced by inner automorphism of $G$)?

thanks a lot!