Locally smooth \(\text{SU}(n+1)\)-actions on \(\text{SU}(n+1)/\text{S}(\text{U}(n-1) \times \text{U}(2))\) are unique
Let \(G\) be a compact connected Lie group and let \(U\subset G\) be a closed connected subgroup. The homogeneous space \(G/U\) is called \textit{standard} if every locally smooth effective action of \(G\) on \(G/U\) is conjugate to the action given by left translation. The author proves that \(\text{SU}(n+1)/\text{S}(\text{U}(n-1) \times \text{U}(2))\) is standard. The key of the proof is the non existence of a set of fixed points for actions of \(\text{SU}(n+1)\). This can be done through the study of equivariant cohomology for actions with principal isotropy group of positive dimension.
equivariant cohomology of homogeneous spaces
scientific article; zbMATH DE number 5118031