different Shimura data with common underlying group?

Question

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!

Answer 1

The question is essentially about ${\mathbf{R}}$-groups, so we shall assume that $G$ is defined over $\mathbf{R}$.

It is not true that for a given connected reductive ${\mathbf{R}}$-group $G$, there are at most finitely many Shimura data of the form $(G,X)$ up to isomorphism. Indeed, assume that $G$ is a compact (i.e. anisotropic) ${\mathbf{R}}$-torus. Then for any homomorphism $h\colon \mathbf{C}^* / \mathbf{R}^* \to G$, the pair $(G,h)$ is a Shimura datum.

Now assume that $G$ is adjoint and simple. I assume that the question is about classification of Shimura data $(G,X)$ up to isomorphism which is the identity on $G$. The possible Shimura data $(G,X)$ are described in Section 1.2 of the paper: P. Deligne, Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques. Automorphic forms, representations and $L$-functions, Part 2, pp. 247–289, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979.

To a $G({\mathbf{R}})$-conjugacy class $X$ of homomorphisms $h \colon \mathbf{C}^* / \mathbf{R}^* \to G$ satisfying conditions 1.2.1(i),(ii),(iii) of the paper, Deligne associates a special vertex $s=s(X)$ of the Dynkin diagram $D$ of $G_{\mathbf{C}}$. In Proposition 1.2.7 he says that the set of all such homomorphisms has two connected components, interchanged by $h\mapsto h^{-1}$. In Corollary 1.2.8 he says:

(i) If $s(X)$ is not fixed by the opposition involution, then $G({\mathbf{R}})$ and $X$ are connected (so I conclude that there exist two such conjugacy classes).

(ii) If $s(X)$ is fixed by the opposition involution, then $G({\mathbf{R}})$ and $X$ have two connected components (so I conclude that there is only one conjugacy class).

Table 1.3.9 shows that there is one conjugacy class for the following adjoint ${\mathbf{R}}$-groups of Hermitian type: $PSU(p,p)$, $B_l$, $C_l$, $D_l^{\mathbf{R}}$, $D_{2l}^{\mathbf{H}}$, $E_7$. Note that for $D_{2l}^{\mathbf{H}}$ the opposition involution is trivial, see e.g. A.L. Onishchik and E.B. Vinberg, Lie Groups and Algebraic Groups, Springer-Verlag 1990, Table 3 on page 298.

EDIT: I answer the second part of the question. Let $G=PGL_{2,\mathbf{R}}$, and let $T\subset G$ be a compact maximal torus. There exists exactly one conjugacy class $X$ of homomorphisms satisfying conditions 1.2.1(i),(ii),(iii) of Deligne's paper. Let $i \colon T \hookrightarrow G$ be the inclusion homomorphism. There exists a homomorphism $h \colon \mathbf{C}^* / \mathbf{R}^* \to T$ such that $i \circ h \in X$. Then $i \circ h^{-1}$ also satisfies conditions 1.2.1(i),(ii),(iii) of Deligne's paper, hence $i \circ h^{-1}\in X$. We see that $(T,h)$ and $(T,h^{-1})$ are different Shimura subdata of $(G,X)$ with the same subgroup $T$.