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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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