A (split) reductive linear algebraic group is equivalently described by combinatorial information called a root datum.
A toric variety is described by combinatorial information called a fan.
Both correspondences use the character lattice.
says that spherical varieties are a nice class of objects that include all my favorite spaces (e.g. symmetric spaces, toric varieties) . And, moreover, that a spherical variety is equivalent to combinatorial information called a colored fan. Is there any way of recovering a root datum from a colored fan? Or is a reductive group actually given as part of the data of a colored fan?
Are fans/ Toric varieties and root data/ reductive groups both special cases of a larger pattern (for example, colored fans/ spherical varieties)?