Let $(G,X)$ be a Shimura datum in the sense of Definition 5.5 of Introduction to Shimura varieties (Milne, 2017). Let $K$ be a compact open subgroup of $G(A_f)$ that is sufficiently small. Let $S$ be a connected component of $\mathrm{Sh}_K(G,X)$. Then is $S$ a connected Shimura variety in the sense of the first half of Definition 4.10 loc.cit.?
It holds when $G^{\mathrm{der}}$ is simply connected by Thm. 5.17 loc.cit. I am curious about the general case.
Yes, it is a connected Shimura variety, but when the derived group is not simply connected, it is more difficult to describe the discrete group $\Gamma$. This is discussed a little in 5.23 of Milne's article. Deligne doesn't assume the derived group is simply connected in his Corvallis article, but that makes things more complicated.