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

Budak, T.; Işık, N.; Milnes, P.; Pym, J.

doi:10.1090/S0002-9939-01-05984-6

The action of a semisimple Lie group on its maximal compact subgroup
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by T. Budak, N. Işık, P. Milnes and J. Pym PDF

Proc. Amer. Math. Soc. 129 (2001), 1525-1534 Request permission

Abstract:

In this paper we determine the structure of the minimal ideal in the enveloping semigroup for the natural action of a connected semisimple Lie group on its maximal 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 $G=\operatorname {SL}(2, \mathbb {C})$ and $G=\operatorname {SL}(2, \mathbb {R}).$

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