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(Disclaimer: this question is related to this question, but is different enough that it warrants (in my opinion) a separate question)

Let $G$ be a connected reductive group over $\mathbb{Q}$. To $G$ and a compact open subgroup $K_f\subseteq G(\mathbb{A}_f)$ there is an associated locally symmetric space

$$S_{K_f}(G):= G(\mathbb{Q})\backslash G(\mathbb{A})/(A_G(\mathbb{R})^+ K_f K_\infty)$$

Here $K_\infty$ is a maximal compact subgroup of $G(\mathbb{R})^+$ (where $+$ denotes the connected component) and $A_G$ is the maximal split torus in $Z(G)$. I think that since any two choices of such a $K_\infty$ are conjugate, there is no dependence on the choice.

My question: How much does the Shimura variety associated to $(G,X)$ depend on $X$ (assuming there exists an $X$)? Here are more specific questions:

  1. How many $(G,X)$ can there be if $G_\mathbb{R}$ is a unitary group $U(p,q)$, a general unitary group $GU(p,q)$, or a general symplectic group $GSp_{2n}$? What are some examples of different choices of $X$ if they exist?
  2. Is $\mathrm{Sh}_{K_f}(G,X)$ (for an arbitary $X$) isomorphic to $S_{K_f}(G)$ as complex analytic spaces equivariant for the Hecke action? What about as real analytic spaces?

Thank you for reading.

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    $\begingroup$ For a real simple adjoint $G$, there are only one or two $X^+$ such that $(G,X^+)$ is a connected Shimura datum. See Deligne's Corvallis article 1.2.7 and 1.2.8. $\endgroup$
    – anon
    Commented Oct 8, 2019 at 13:22

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