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The Weyl group as fixed point set of smooth involutions. (English) Zbl 0903.22005
As usual, write $$G=\text{KAN}$$ for the Iwasawa decomposition of a semisimple connected non-compact Lie group $$G$$ with finite centre. Denote by $$M$$ and $$M'$$ respectively, the centralizer and normalizer in $$K$$ of the Lie algebra of $$A$$. The Weyl group $$W=M'/M$$ may be regarded as a finite subset of $$K/M$$. It is easy to exhibit $$W$$ as the fixed point set of a single discontinuous involution but looking at a few basic examples like $${\text{SL}}(2,{\mathbb R})$$, $${\text{SL}}(3,{\mathbb R})$$, and $${\text{SU}}(2,1)$$ indicates that it might be possible to see $$W$$ as the common fixed points of a set of smooth involutions. As is proved in this article, this is true in general. In fact, a careful analysis of the $${\text{SU}}(2,1)$$ case provides the key to the general case via the Bruhat decomposition of $$K$$. It is a rather natural result which will surely find further application.

##### MSC:
 22E46 Semisimple Lie groups and their representations 43A85 Harmonic analysis on homogeneous spaces
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