On the Classification of Transitive Effective Actions on Stiefel Manifolds

Introduction. Let G be a compact connected Lie group and H a closed subgroup of G. Then the coset space G/H is a smooth manifold with G acting transitively as translations. It is easy to see that the natural action of G on G/H is effective if and only if H contains no nontrivial normal subgroup of G. For a given compact homogeneous space M=G/H, one might ask whether there are any other (differentiably nonequivalent) transitive effective actions on M? And furthermore, what are all the possible nonequivalent transitive effective actions on M ? In the special case that M is a sphere, the above classification problem has been completely solved by the successive efforts of Montgomery and Samelson [10], Borel [1], and Poncét [11]. The purpose of this paper is to continue, along this direction, to classify the transitive effective actions on the Stiefel manifolds. Let Vn.k be the Stiefel manifold of orthonormal (« — fc)-frames in the euclidean «-space. If we consider Vn.k as a subset of the space of nx(n—k) matrices, then SO(«) acts on Vn.k by matrix multiplication from the left and SO(«— k) acts on Vn¡k by matrix multiplication from the right. Suppose G is any compact Lie group such that SO(«)cG<=SO(«) x SO(«-Zr), then it is easy to see that G acts on Vn.k transitively (and effectively in many cases). Our main result is that: "For many values of « and k, every transitive effective action on Vn.k is differentiably equivalent to one of the above examples." Parallel results are also proved for the complex and symplectic Stiefel manifolds. Technically, the most difficult part of the proof is to show that any transitive group G on Vn.k contains a simple normal subgroup Gx that already acts transitively! In the case of transitive groups on spheres, the above fact is an easy consequence of the particularly simple structure of the cohomology group of spheres [10], which is no longer available in our case. For this purpose we introduce the concept of irreducible transitive action, namely, G is said to be an irreducible transitive group on M if there is no proper normal subgroup of G that is already transitive on M. With the help of a cohomological criterion of irreducible transitivity, the uniqueness of irreducible transitive effective action on a Stiefel manifold is established