R-spaces associated with a Hermitian symmetric pair

Abstract

The linear isotropy representation of a Riemannian symmetric pair (G,K) is defined as the differentialof the left action of K on GJK at the origin. Every orbit of the linear isotropy representation of (G,K) is called an R-space associated with {G,K), which is an important example of equivariant homogeneous Riemannian submanifolds in a Euclidean sphere (See Takagi-Takahashi [2] and TakeuchiKobayashi [3]). This paper is concerned with the linear isotropy representation of a Hermitian symmetric pair (G,K). Its restriction to the center of K defines an S1-action on the associated i?-spaces. We determine all i?-spaces associated with Hermitian symmetric pairs (G,K) on which the semisimple part of K acts transitively. In particular,we know allirreducible Hermitian symmetric pairs such that each of the associated i?-spaces has such a property. This result is utilizablefor the classificationof orthogonal transformation groups by their cohomogeneity (See the forthcoming paper [4] concerned with this problem in low cohomogeneity). The authors are profoundly grateful to Professor Ryoichi Takagi for his helpful suggestion and criticalreading of a primary manuscript.