Groups of projectivities and Levi subgroups in spherical buildings of simply laced type

Abstract We introduce the special and general projectivity groups attached to a simplex 𝐹 of a thick, irreducible, spherical building of simply laced type. If the residue of 𝐹 is irreducible, we determine the permutation group of both projectivity groups of 𝐹, acting on the residue of 𝐹 and show that the special projectivity group determines the precise action of the Levi subgroup of a parabolic subgroup on the corresponding residue. This reveals three special cases for the exceptional types