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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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