Relationship between fans and root data

(1) A (connected) reductive group $G$ over an algebraically closed field $k$ is described by a combinatorial object called the based root datum ${\rm BRD}(G)$.

(2) A spherical homogeneous space $Y=G/H$ is a homogeneous space on which a Borel subgroup $B$ of $G$ acts with an open Zariski-dense orbit. It is described (uniquely at least in characteristic 0) by its homogeneous combinatorial invariants. These combinatorial invariants constitute an additional structure on ${\rm BRD}(G)$.

(3) A spherical embedding $G/H\hookrightarrow Y^e$ is a normal $G$-variety $Y^e$ containing a spherical homogeneous space $G/H$ as an open dense $G$-orbit. It is described by its colored fan, which is an additional structure on the homogeneous combinatorial invariants.

By spherical varieties one means spherical homogeneous spaces and spherical embeddings.

Therefore, I think that the based root datum of $G$ should be regarded as a part of data describing the $G$-variety $Y^e$.

In the case when $G=T$ is a torus, we take $H=1$, and then the spherical embeddings of $G/H=T$ are the same as the toric varieties for $T$, and the corresponding colored fans are just fans.

Reference: Nicolas Perrin, On the geometry of spherical varieties.

Not an answer, but: you can construct a fan from a root system. Let $R$ be a root system in an Euclidean space, and let $\Lambda_R$ be the root lattice with dual lattice $\Lambda_R^\vee$. The fan $\Sigma$ in $\Lambda_R^\vee$ associated to $R$ consists of the Weyl chambers of $R$ and all their faces. For instance, if $R=A_1$, then the associated toric variety is $\mathbf{P}^1$. I don't know how to determine when a fan comes from a root system, but I'm guessing someone here does.