De Mari, Filippo The Weyl group as fixed point set of smooth involutions. (English) Zbl 0903.22005 J. Lie Theory 6, No. 1, 53-68 (1996). 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. Reviewer: M.G.Eastwood (Adelaide) Cited in 1 Document MSC: 22E46 Semisimple Lie groups and their representations 43A85 Harmonic analysis on homogeneous spaces Keywords:Lie group; Weyl group; involutions; Iwasawa decomposition; semisimple connected non-compact Lie group; fixed point set PDFBibTeX XMLCite \textit{F. De Mari}, J. Lie Theory 6, No. 1, 53--68 (1996; Zbl 0903.22005) Full Text: EuDML