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Polar coordinates induced by actions of compact Lie groups. (English) Zbl 0565.22010
Let G be a compact Lie group acting on a real vector-space V. The author constructs linear subspaces \({\mathfrak a}\subset V\) that intersect all G- orbits. The most interesting situation arises when the G-orbits are orthogonal to \({\mathfrak a}\), but in this case it is natural to think of \({\mathfrak a}\) and the G-orbits as giving polar coordinates on V, in analogy with the action of SO(n) on \({\mathbb{R}}^ n\). Therefore the real representations of G, whose orbits admit orthogonal linear cross-sections are called polar. It is proved that all polar coordinates on real vector- spaces induced by actions of compact Lie groups can be obtained from the symmetric space actions. Moreover the fundamental property of polar representations and the classification of all real irreducible polar representations are given.
Reviewer: A.Fleischer

22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
57S15 Compact Lie groups of differentiable transformations
Full Text: DOI
[1] Jiri Dadok and Victor Kac, Polar representations, J. Algebra 92 (1985), no. 2, 504 – 524. · Zbl 0611.22009
[2] Jiri Dadok and Reese Harvey, Calibrations on \?\(^{6}\), Duke Math. J. 50 (1983), no. 4, 1231 – 1243. · Zbl 0535.49030
[3] SigurĂ„’ur Helgason, Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962.
[4] Sigurdur Helgason, Analysis on Lie groups and homogeneous spaces, American Mathematical Society, Providence, R.I., 1972. Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 14. · Zbl 0264.22010
[5] James E. Humphreys, Introduction to Lie algebras and representation theory, Springer-Verlag, New York-Berlin, 1972. Graduate Texts in Mathematics, Vol. 9. · Zbl 0254.17004
[6] G. A. Hunt, A theorem of Elie Cartan, Proc. Amer. Math. Soc. 7 (1956), 307 – 308. · Zbl 0073.01602
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