Iwasawa decompositions of groups with a root group datum

The Iwasawa decomposition of a connected semisimple complex Lie group or a connected semisimple split real Lie group is one of the most fundamental observations of classical Lie theory. It implies that the geometry of a connected semisimple complex resp. split real Lie group G is controlled by any maximal compact subgroup K. Examples are Weyl’s unitarian trick in the representation theory of Lie groups, or the transitive action of K on the Tits building G/B. In the case of the connected semisimple split real Lie group of type G2 the latter implies the existence of an interesting epimorphism from the real building of type G2, the split Cayley hexagon, onto the real building of type A2, the real projective plane, by means of the epimorphism SO4(R) → SO3(R), cf. [14]. This epimorphism cannot be described using the group of type G2, because it is quasisimple.