$p$-local decompositions of projective Stiefel manifolds

The main objective of this paper is to analyze the $p$-local homotopy type of the complex projective Stiefel manifolds, and other analogous quotients of Stiefel manifolds. We take the cue from a result of Yamaguchi about the $p$-regularity of the complex Stiefel manifolds which lays down some hypotheses under which the Stiefel manifold is $p$-locally a product of odd dimensional spheres. We show that in many cases, the projective Stiefel manifolds are $p$-locally a product of a complex projective space and some odd dimensional spheres. As an application, we prove that in these cases, the $p$-regularity result of Yamaguchi is also $S^1$-equivariant.