Homogeneous quaternionic Kähler structures and quaternionic hyperbolic space

Abstract

Homogeneous Riemannian structures have been studied and classified in terms of tensors through the works of Ambrose-Singer and of Tricerri-Vanhecke, dating back to the 1950s and 1980s, respectively. More recently, an abstract representation theoretic decomposition of the space V of tensors satisfying the symmetries of a homogeneous Riemannian structure has been proposed by A. Fino [Math. J. Toyama Univ. 21 (1998), 1–22; ] in the context of H-homogeneous structures, H being any of the possible irreducible holonomy groups. The paper under review deals with homogeneous quaternionic Kähler structures and its first result is a concrete description of the decomposition of V into five basic subspaces QK1,…,QK5 invariant under the action of Sp(n)⋅Sp(1), n≥2. Besides this decomposition, the main statements, anticipated in the note [M. Castrillón López, P. M. Gadea and A. Swann, C. R. Math. Acad. Sci. Paris 338 (2004), no. 1, 65–70; ], concern homogeneous quaternionic Kähler structures on the quaternionic hyperbolic space HHn. It is shown in particular that all such structures are in the class QK3 and that they are realized by the homogeneous models Sp(1)RN/Sp(1), where N is the nilpotent factor in the Iwasawa decomposition of Sp(n,1) and the isotropy representation depends on a positive real parameter.