The action of a semisimple Lie group on its maximal compact subgroup (Q2701654)

The authors determine the structure of the minimal ideal in the enveloping semigroup for the natural action of a connected semisimple Lie group on its maximal connected compact subgroup. In particular, if \(G= KAN\) is an Iwasawa decomposition of the group \(G\), then the group in the minimal left ideal is isomorphic both algebraically and topologically with the normalizer \(M\) of \(AN\) in \(K\). Complete descriptions are given for the enveloping semigroups in the cases \(G= \text{SL}(2,\mathbb{C})\) and \(G= \text{SL}(2, \mathbb{R})\).