On actions of adjoint type on complex Stiefel manifolds

Let $G(m)$ denote $\mathrm{S}\mathrm{U}(m)$ or $\mathrm{S}\mathrm{p}(m)$ . It is shown that when $m\ge 5G(m)$ cannot act smoothly on $W_{n,2}$ , the complex Stiefel manifold of orthonormal $2$ -frames in ${\mathbf{C}}^n$ , for $n$ odd, with connected principal isotropy type equal to the class of maximal tori in $G(m)$ . This demonstrates an important difference between $W_{n,2}$ , $n$ odd, and $S^{2n-3}\times S^{2n-1}$ in the behavior of differentiable transformation groups. Exactly the same holds for $\mathrm{S}\mathrm{O}(m)$ or Spin $(m)$ if it is further assumed that a maximal $2$ -torus of $\mathrm{S}\mathrm{O}(m)$ has fixed points. ${}^2$