Examples of continuous semisimple Lie group actions not equivalent to smooth ones (Q1966202)

One aspect of Hilbert's fifth problem is the question if every continuous local action of a local Lie group is locally equivalent to a smooth action. For global actions and global equivalence, this was answered in the negative by \textit{G. E. Bredon} [Introduction to compact transformation groups, Pure Appl. Math. Academic Press, Vol. 46 (1972; Zbl 0246.57017)]. The author gives a negative answer to the original local problem. He takes any smooth effective action on the \(n\)-sphere of a connected semisimple Lie group \(G\) without compact quotients. If a maximal compact subgroup is transitive on all \(G\)-orbits, then the cone and the suspension of the \(G\)-action are not locally equivalent at the fixed points to any smooth local action. Concrete examples are given where \(G\) is noncompact simple and a maximal compact subgroup is transitive on the sphere, namely, conformal groups and the effective coverings of the action of \(\text{PSL}_2\mathbb{R}\) on the projective line (which is the conformal group of the circle).