Why do we ask Shimura datum to have Hodge weight $(-1,1),(0,0),(1,-1)$?
I know it's related to the decomposition of a complex Lie algebra $\frak{g}_{\mathbb{C}}=\frak{t}\oplus\frak{p}^{+} \oplus \frak {p} ^{-}$ but I'm having trouble finding what in Shimura's varieties theory needs this assumption.
Does anyone have an enlightening explanation?
This is the condition which gives the Shimura variety the (almost) complex structure, which is obviously necessary if you want to view it as a variety over $\mathbb C$. This is explained in Theorem 1.21 of Milne's notes (in slightly different language, but all the equivalences are established later in the text), but let me summarize here.
Let $(G,X)$ be a Shimura datum, where $X$ is the conjugacy class of maps $h:\mathbb S\to G_{\mathbb R}$. At the level of real points, $h$ gives a map $h:\mathbb C^\times\to G(\mathbb R)$, which via the adjoint action gives an action of $\mathbb C^\times$ on the tangent space of $G(\mathbb R)$ at the identity. The subspace on which this subgroup is trivial coincides with the subspace tangent to the compact subgroup $K$ stabilizing $h$, which gives an action of $\mathbb C^\times$ on the tangent space $T_hX$. Now the condition on weights comes in, and tells us that the only characters of $\mathbb C^\times$ occurring in the complexification $(T_hX)_{\mathbb C}$ are $z$ and $\bar z$, and $T_hX$ can be identified with the subspace of the complexification on which the character is $z$. This equips $T_hX$ with a complex vector space structure, and by homogeneity, $X$ receives an almost complex structure.
