Automorphisms and opposition in spherical buildings of exceptional type, I

Parkinson, James ; Maldeghem, Hendrik Van

Canadian journal of mathematics, Tome 74 (2022) no. 6, pp. 1517-1578

Résumé

To each automorphism of a spherical building, there is a naturally associated opposition diagram, which encodes the types of the simplices of the building that are mapped onto opposite simplices. If no chamber (that is, no maximal simplex) of the building is mapped onto an opposite chamber, then the automorphism is called domestic. In this paper, we give the complete classification of domestic automorphisms of split spherical buildings of types $\mathsf {E}_6$, $\mathsf {F}_4$, and $\mathsf {G}_2$. Moreover, for all split spherical buildings of exceptional type, we classify (i) the domestic homologies, (ii) the opposition diagrams arising from elements of the standard unipotent subgroup of the Chevalley group, and (iii) the automorphisms with opposition diagrams with at most two distinguished orbits encircled. Our results provide unexpected characterizations of long root elations and products of perpendicular long root elations in long root geometries, and analogues of the density theorem for connected linear algebraic groups in the setting of Chevalley groups over arbitrary fields.

Détail

DOI : 10.4153/S0008414X21000341

Mots-clés : Exceptional spherical buildings, opposition diagram, domestic automorphism