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

Straume, Eldar

doi:10.1090/S0002-9947-1988-0927687-8

Weyl groups and the regularity properties of certain compact Lie group actions
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Trans. Amer. Math. Soc. 306 (1988), 165-190 Request permission

Abstract:

The geometric weight system of a $G$-manifold $X$ (acyclic or spherical) is the nonlinear analogue of the weight system of a linear representation. We study the possible realization of a given $G$-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 $G$-weight pattern must in fact coincide with that of a simple representation. This in turn implies that $X$ is (orthogonally) modeled on the linear $G$-space, e.g., with the same orbit types. In particular, complete results in this direction are obtained for a certain family of $G$-manifolds, $G$ a classical group. In this family the weight patterns are of $2$-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.

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