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

We determine the exact structure and action of Levi subgroups of parabolic subgroups of groups of Lie type related to thick, irreducible, spherical buildings of simply laced type. Therefore we introduce the special and general projectivity groups attached to simplices $F$. If the residue of $F$ is irreducible, we determine the permutation group of the projectivity groups of $F$ acting on the residue of $F$ and show that this 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 $E_6$, $E_7$, $E_8$. Furthermore, we establish a general diagrammatic rule to decide when exactly the special and general projectivity groups of $F$ coincide.