The definition of a "connected Shimura datum" (as in Milne's notes) is a pair $(G, X)$, where $G$ is a reductive algebraic group and $X$ is a $G(\mathbb{R})$-conjugacy class of morphisms
$$
x: \mathbb{S}^1 \to G\_\mathbb{R},
$$
where $\mathbb{S}^1$ is the norm one subtorus of $\text{Res}\_{\mathbb{C}/\mathbb{R}}$ satisfying a short list of axioms. Given such a morphism $x$, one gets a family of morphisms
$$
x\_n: \text{Res}\_{\mathbb{C}/\mathbb{R}} \mu\_n \to G\_\mathbb{R}
$$
compatible with the natural inclusions $\mu\_n \hookrightarrow \mu\_{mn}$, and if I'm not mistaken, by Zariski density, $x$ is determined uniquely by the $x\_n$, and moreover, two maps $x$, $x'$ are $G(\mathbb{R})$ conjugate if and only if their associated families $x\_n$, $x\_n'$ are. It also makes sense to demand that Deligne's axioms hold for the $x\_n$, and one sees that if they hold for $x$, they hold for $x\_n$.


From the perspective of special points and canonical models, it is not clear to me where one uses the morphism $x$; on the level of real points, it seems that only the action of roots of unity are used, and so much of the theory should be recoverable from only the $x\_n$; however, there is a lot of it that I haven't understood yet.


My questions are:


1. if we define a generalized Shimura datum to be a family of $x\_n$ compatible with the natural inclusions (equivalently, a conjugacy class of maps from the direct limit of the $\mu\_n$), do we get generalized Shimura varieties?
2. If so, are there generalized Shimura varieties which do not come from Shimura varieties?



What is wrong with this modification of the definition of Shimura datum?
Any reasonable interpetation of the first axiom (on the weights of the action on the adjoint representation) will force the homomorphisms $\mu\_n \to G$ to canonically factor through $\mathbb G\_m$, and so a generalized Shimura variety is just a Shimura variety.


The reason is that any representation of $\mu\_\infty$ is a sum of one-dimensional representations. One-dimensional representations are classified by $\hat{\mathbb Z}$, and the ones that factor through $\mathbb Z$ are classified by $\mathbb Z$. The ones appearing in the adjoint representation lie in $\{-1,0,+1\}$, hence in $\mathbb Z$. Since a tensor power of any representation is contained in a tensor power of the adjoint representation, a power of any character appearing in the representation will be classified by $\mathbb Z$, so every character will be classified by an element of $\mathbb Q$, but $\mathbb Q \cap \hat{\mathbb Z}= \mathbb Z$.


