General and Special Projectivity Groups in Spherical Buildings of Simply Laced Type

We introduce the special and general projectivity groups attached to simplices F of spherical buildings and study those in the simply laced irreducible thick case. If the residue of F is irreducible, we determine the permutation group of the projectivity groups of F acting on the residue of F. This pins down the exact structure and action of the Levi complement of the corresponding parabolic subgroup in the simple Chevalley group, revealing three special cases for the exceptional types E_6, E_7 and E_8. Furthermore, we establish a general diagrammatic rule to decide when exactly the special and general projectivity groups of F coincide.