Definition for Shimura datum The following definition for $\textbf{shimura datum}$ is due to wikipedia.
Let $S=\mathrm{Res}_\mathbb{R}^\mathbb{C}G_m$ be the Weil restriction of the multiplicative group from complex field $\mathbb{C}$ to real field $\mathbb{R}$. A $\textbf{shimura datum}$ is a pair $(G,X)$ consisting of a reductive algebraic group $G$ defined over the rational number field $\mathbb{Q}$ and a $G(\mathbb{R})$-conjugacy class $X$ of homomorphisms $h:S\rightarrow G_\mathbb{R}$ satisfying the following axioms:
(i) The complexified Lie algebra of $G$ decomposes into a direct sum $\mathfrak{g}\bigotimes\mathbb{C}=\mathfrak{k}\bigoplus\mathfrak{p}^+\bigoplus\mathfrak{p}^-$, where for any $z\in S$, $h(z)$ acts trivially on the first summand and via $\frac{z}{\bar{z}}$ (respectively, $\frac{\bar{z}}{z}$ on the second (respectively, third) summand.
(ii) The adjoint action of $h(i)$ induces a Cartan involution on the adjoint group of $G_\mathbb{R}$.
(iii) The adjoint group of $G_\mathbb{R}$ does not admit a factor $H$ defined over $\mathbb{Q}$ such that the projection of $h$ on $H$ is trivial.
It is not quite clear to me in this definition.
(a) If $g\in G(\mathbb{R})$ and $h:S\rightarrow G_\mathbb{R}$, how does $g$ act $h$? Is it given by $(g\cdot h)(z):=g^{-1}h(z)g$?
(b) In (i), what does "$h(z)$ acts via $\frac{z}{\bar{z}}$" mean? Does $h(z)$ act as multiplying by $\frac{z}{\bar{z}}$?
(c) In (iii), What is "a factor H"? Is $H$ a subgroup? Then what is "the projection of $h$ on H"?
 A: (a) No, the action is by conjugation--if one has a homomorphism $\mathbb{S}\to G_\mathbb{R}$ one can conjugate this by any elementof $G(\mathbb{R})$ to obtain another such homomorphism.
(b) The map $\mathbb{S}\to G_\mathbb{R}$ induces a representation
$$\mathbb{S}_\mathbb{C}\to \mathrm{GL}(\mathfrak{g}_\mathbb{C})$$
in the obvious way. But, the group $\mathbb{S}_\mathbb{C}\cong \mathbb{G}_m\times \mathbb{G}_m$. Thus, we get a decomposition
$$\mathfrak{g}_\mathbb{C}=\bigoplus_{p,q}V^{p,q}$$
where $(z_1,z_2)$ in $\mathbb{C}^\times\times\mathbb{C}^\times$ acts on $V^{p,q}$ by $z_1^p z_2^q$. It's really just saying that it wants $V^{p,q}$ to be non-zero for $(p,q)$ in $\{(-1,1),(0,0),(1,-1)\}$.
If you want to read more about this google 'Deligne torus' and 'Hodge structure'. If you'd like to hear more about the geometric reason for this assumption, let me know.
(c) The group $G^\mathrm{ad}$, being adjoint, is actually isomorphic to a product of simple $\mathbb{Q}$-groups $G_1\times \cdots \times G_n$, which the $G_i$ are just the simple normal subgroups of $G^\mathrm{ad}$ (e.g. [Mil, §24.a]). Condition (c) is then just saying that the composition
$$h\to G_\mathbb{R}\to G^\mathrm{ad}_\mathbb{R}\to (G_i)_{\mathbb{R}}$$
is non-trivial for all $i$.
[Mil Milne, J.S., 2017. Algebraic groups: The theory of group schemes of finite type over a field (Vol. 170). Cambridge University Press.
