On eigenvalues on the orbit space

A map P equivariant with respect to a compact Lie group induces a map Q on the orbit space, which is differentiable provided that P is sufficiently smooth. An equilibrium of Q corresponds to an invariant orbit of P. We compare the eigenvalues of the linearization at the equilibrium with the eigenvalues of a kind of linearization at the invariant orbit. It turns out that the former are products of the latter. Furthermore, the eigenvalues on the orbit space suffice to determine asymptotic stability or instability of the invariant orbit. This generalizes a similar result for vector fields by Kœnig, Math. Proc. Cambridge, Philos. Soc. 121 (1997) 401–424 and applies also to relative periodic points.