R-spaces associated with a Hermitian symmetric pair (Q1088166)

As in the case of Riemannian symmetric pairs, the orbit of the linear isotropy representation of a Hermitian symmetric pair (G,K) is called an R-space associated with (G,K). In this paper te authors give a characterization of those R-spaces on which the semi-simple part \(K_ s\) of the group K acts transitively. Let \({\mathfrak g}\) and \({\mathfrak k}\) be the Lie algebras of G and K, respectively. Then \({\mathfrak g}={\mathfrak k}+{\mathfrak m}\) (direct sum). Denote by \({\mathfrak a}^ c\) the complexification of a maximal abelian subspace of \({\mathfrak m}\), and consider the set R of all non-zero elements in the root- space decomposition of \({\mathfrak g}^ c\), the complexification of \({\mathfrak g}\), with respect to the Cayley transform v of \({\mathfrak a}^ c\). Harish- Chandra has then shown that there are only two possibilities in R and called them type C and type BC. Utilizing this fact, it is proved here that \(K_ s\) acts transitively on each of the associated R-spaces if and only if R is of type BC.