Equivariant cross sections of complex Stiefel manifolds

Let G be a finite group and let M be a unitary representation space of G. A solution to the existence problem of G-equivariant cross sections of the complex Stiefel manifold Wk(M) of unitary k-frames over the unit sphere S(M) is given under mild restrictions on G and on fixed point sets. In the case G is an even ordered group, some sufficient conditions for the existence of G-equivariant real frame fields on spheres with complementary G-equivariant complex structures are also obtained, improving earlier results about odd ordered groups.