On actions of regular type on complex Stiefel manifolds

The usual unitary representations of the special unitary, symplectic, or special orthogonal groups define a sequence of smooth actions on the complex Stiefel manifolds called the regular linear models. If one of the above groups acts smoothly on the complex Stiefel manifold of orthonormal $2$ -frames in ${\mathbf{C}}^n$ for odd $n$ , and if the identity component of the principal isotropy type is of regular type, then it is shown under mild dimension restrictions that the orbit structure and the cohomology structure of the fixed point varieties (over the $\mathrm{mod}2$ Steenrod algebra) resemble those of the regular linear models. The resemblance is complete in the cases of the special unitary and symplectic groups. There is an obstruction to complete resemblance in the case of the special orthogonal groups. An application of the above regularity theorems is given.