Groups of Projectivities and Levi Subgroups in Spherical Buildings of Simply Laced Type

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 ${\mathsf{E}}_{\mathsf{6}},{\mathsf{E}}_{\mathsf{7}},{\mathsf{E}}_{\mathsf{8}}$. Furthermore, we establish a general diagrammatic rule to decide when exactly the special and general projectivity groups of $F$ coincide.