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A generalization of the Weyl group. (English) Zbl 0545.57014
Let G be a compact connected Lie group of isometries of a complete Riemannian manifold M such that all isotropy groups \(G_ x\) are of maximal rank. Choose a principal orbit G(x) and let \(F_ x\) be the component of x in the fixed point set of \(G_ x\). The author proves that the union S of all singular orbits is equal to the strong focal locus of G(x). Moreover, the intersection \(S\cap F_ x\) is of codimension 1 in \(F_ x\) and consists of a submanifold of codimension 1, with components \(C_ 1,...,C_ q\), plus a set of ’branch points’. Each \(C_ i\) is contained in the fixed point set \(F_ i\) of an involution \(g_ i\in N(G_ x)/G_ x\) acting on \(F_ x\). The ’walls’ \(F_ i\) decompose \(F_ x\) into ’generalized Weyl chambers’, and the involutions \(g_ i\) generate the ’generalized Weyl group’ of the action.
In the special case where M is the Lie algebra of G and the action is the adjoint action, \(F_ x\) becomes a Cartan subalgebra and the construction above reduces to a description of the ordinary Weyl group and Weyl chambers as given by W. Y. Hsiang [Cohomology theory of topological transformation groups (1975; Zbl 0429.57011)].
Reviewer: R.Löwen

MSC:
57S15 Compact Lie groups of differentiable transformations
53C20 Global Riemannian geometry, including pinching
22E15 General properties and structure of real Lie groups
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