On actions of adjoint type on complex Stiefel manifolds

Wang, McKenzie Y.

doi:10.1090/S0002-9947-1982-0662056-4

On actions of adjoint type on complex Stiefel manifolds
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Trans. Amer. Math. Soc. 272 (1982), 611-628 Request permission

Abstract:

Let $G(m)$ denote ${\rm {SU}}(m)$ or ${\rm {Sp}}(m)$. It is shown that when $m \geq 5 G(m)$ 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 ${\rm {SO}}(m)$ or Spin$(m)$ if it is further assumed that a maximal $2$-torus of ${\rm {SO}}(m)$ has fixed points.$^{2}$

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