Weyl groups and the regularity properties of certain compact Lie group actions

Author:
Eldar Straume

Journal:
Trans. Amer. Math. Soc. **306** (1988), 165-190

MSC:
Primary 57S15; Secondary 20G05

DOI:
https://doi.org/10.1090/S0002-9947-1988-0927687-8

MathSciNet review:
927687

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Abstract: The geometric weight system of a -manifold (acyclic or spherical) is the nonlinear analogue of the weight system of a linear representation. We study the possible realization of a given -weight pattern, via the interaction between roots, weights and the Weyl group, together with various fixed point results of P. A. Smith type. If the orbit structure is reasonably simple, then the -weight pattern must in fact coincide with that of a simple representation. This in turn implies that is (orthogonally) modeled on the linear -space, e.g., with the same orbit types. In particular, complete results in this direction are obtained for a certain family of -manifolds, a classical group. In this family the weight patterns are of -parametric type, and it includes essentially all cases where the principal isotropy type is nontrivial. This family also covers many cases with trivial principal isotropy type.

**[1]**A. Borel et al.,*Seminar on transformation groups*, Ann. of Math. Studies, no. 46, Princeton Univ. Press, Princeton, N.J., 1961. MR**0116341 (22:7129)****[2]**N. Bourbaki,*Groupes et algèbres de Lie*, Chapitres IV-VI, Hermann, Paris, 1968. MR**0240238 (39:1590)****[3]**W. C. Hsiang and W. Y. Hsiang,*Differentiable actions of compact connected classical groups*, I, II, III, Amer. J. Math.**89**(1967), 705-786; Ann. of Math.**92**(1970), 189-223; ibid.**99**(1974), 220-256. MR**0217213 (36:304)****[4]**W. Y. Hsiang,*On the splitting principle and the geometric weight system of topological transformation groups*, I, Proc. 2nd Conf. on Compact Transformation Groups, Amherst, Mass., 1971, Lecture Notes in Math., vol. 298, Springer-Verlag, 1972, pp. 334-402. MR**0380845 (52:1742)****[5]**-,*Cohomology theory of topological transformation groups*, Springer-Verlag, 1975. MR**0423384 (54:11363)****[6]**W. Y. Hsiang and E. Straume,*Actions of compact connected Lie groups with few orbit types*, J. Reine Angew. Math.**334**(1982), 1-26. MR**667447 (83m:57032)****[7]**-,*Actions of compact connected Lie groups on acyclic manifolds with low dimensional orbit space*, J. Reine Angew. Math.**369**(1986), 21-39. MR**850627 (87m:57041)****[8]**-,*On the orbit structure of*-*actions on manifolds of the type of euclidean, spherical, or projective spaces*: Math. Ann.**278**(1987), 71-97. MR**909218 (89a:57056)****[9]**E. Straume, -*weights and their application to regular actions of classical groups*, Math. Rep., Univ. of Oslo, 1975.**[10]**-,*Dihedral transformation groups on homology speheres*, J. Pure Appl. Algebra**21**(1981), 51-74. MR**609271 (82h:57035)****[11]**-,*The topological version of groups generated by reflections*, Math. Z.**176**(1981), 429-446. MR**610222 (82d:20047)****[12]**P. Yang,*Adjoint type representations of classical groups on homology spheres*, Thesis, Princeton Univ., Princeton, N.J., 1974.

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DOI:
https://doi.org/10.1090/S0002-9947-1988-0927687-8

Article copyright:
© Copyright 1988
American Mathematical Society