Lie groups transitive on Grassmann and Stiefel manifolds

A. L. Onishchik

Sbornik. Mathematics, Tome 12 (1970) no. 3, pp. 405-427

Résumé

Lie groups which ate transitive on real Grassmann manifolds $G_{n,2k}$ and on quaternionic Grassmann manifolds $Q_{n,k}$ are studied. The principal result states that each connected Lie group which acts transitively and effectively on $G_{n,2k}$ ($22k$) or on $Q_{n,k}$ ($2$) is similar to the real linear group $SL(n,\mathbf R)$ or the quaternionic group $SU^*(2n)$ or their subgroups $SO(n)$ and $Sp(n)$ respectively. The analogous statement for complex Grassmann manifolds was shown by the author previously (Math. Sb. (N.S.) 75(117) (1968), 255–263). Also treated are all simple compact Lie groups which are transitive on real, complex or quaternionic Stiefel manifolds (with some exceptions). From this is obtained a classification of all noncompact simple Lie groups which are transitive on these manifolds and whose maximal compact subgroups contain a unique simple normal divisor of rank greater than 1. Bibliography: 15 titles.