Two-step nilpotent Lie groups and homogeneous fiber bundles (Q1404113)

A new subclass of two-step nilpotent Lie groups is introduced. This class is attached to homogeneous fiber bundles \(H\slash K\to G\slash K\to G\slash H,\) where \((G,H)\) is a compact symmetric pair and \(K\) is a closed subgroup of \(H.\) If \(\mathfrak{g},\) \(\mathfrak{h}\) and \(\mathfrak{k}\) are the Lie algebras of \(G,\) \(H,\) and \(K,\) respectively, then the author constructs the two-step nilpotent Lie algebras \(\mathfrak{n}(\mathfrak{g},\mathfrak{h},\mathfrak{k})\) and investigates its properties. In this paper the author considers the case when \((G,H)\) is irreducible and shows close relations between the isotropy representations of \(G\slash K\) and a simply connected nilpotent Lie group \(N(\mathfrak{g},\mathfrak{h},\mathfrak{k})\) corresponding to \(\mathfrak{n}(\mathfrak{g},\mathfrak{h},\mathfrak{k}).\) The relation between weak symmetry on the total space \(G\slash K\) and the nilpotent groups \(N(\mathfrak{g},\mathfrak{h},\mathfrak{k}),\) found by the author, provides new examples of weakly symmetric nilpotent groups. The author has also found the relation between the geodesic orbit property of \(G\slash K\) and \(N(\mathfrak{g},\mathfrak{h},\mathfrak{k})\). (Recall, that a connected Riemannian manifold has geodesic orbit property if every geodesic is an orbit of a one parameter subgroup of the isometry group.)