Actions of some simple compact Lie groups on themselves

Let G be a compact connected simple Lie group acting non-transitively, non-trivially on itself. Hsiang (Cohomology theory of topological transformation groups, (1975) (New York: Springer)) conjectured that the principal isotropy subgroup type must be the maximal torus and the action must be cohomologically similar to the adjoint action and the orbit space must be a simplex. But Bredon (Bull AMS83(4) (1977) 711-718) found a simple counterexample, where the principal isotropy subgroup is not a maximal torus and which has no fixed point. In this work, we prove that if SO(n), (n >= 34) or SU(3) acts smoothly (and nontrivially) on itself with non-empty fixed point set, then the principal isotropy subgroups are maximal tori.