Abstract involutions of algebraic groups and of Kac-Moody groups

Based on the second author's thesis in this article we provide a uniform treatment of abstract involutions of algebraic groups and of Kac-Moody groups using twin buildings, RGD systems, and twisted involutions of Coxeter groups. Notably we simultaneously generalize the double coset decompositions established by Springer and by Helminck-Wang for algebraic groups and by Kac-Wang for certain Kac-Moody groups, we analyze the filtration studied by Devillers-Muhlherr in the context of arbitrary involutions, and we answer a structural question on the combinatorics of involutions of twin buildings raised by Bennett-Gramlich-Hoffman-Shpectorov

Iwasawa decompositions of split Kac-Moody groups

We characterize all fields F for which a group with an F-locally split root group datum admits an Iwasawa decomposition. This class of groups in particular includes the split semisimple algebraic groups and the split Kac-Moody groups

Moufang sets of type F-4

We give an explicit description of the Moufang sets of type F-4, i.e. the buildings arising from the simple algebraic groups of absolute type F4 and relative rank one, over an arbitrary field. We use octonion planes and certain polarities to find this description, and we rely on the theory of Albert algebras. We also determine the automorphism groups of the corresponding exceptional unitals, thereby completing the program of J. Tits for these non-abelian Moufang sets. In particular we prove that every automorphism of that unital is induced by a collineation of the ambient projective plane