Isometry groups of homogeneous quaternionic Kähler manifolds (Q1589364)

The authors present a general method for calculating the full isometry group of a Riemannian solvmanifold. They then use this method for calculating the full isometry group of the non-symmetric quaternionic Kähler solvmanifolds, that is, non-symmetric quaternionic Kähler manifolds admitting a transitive splittable solvable group of isometries. This is done explicitly and uses the classification of the quaternionic Kähler solvmanifolds. A remarkable consequence is that the connected component of the isometry group of a quaternionic Kähler solvmanifold acts transitively on the twistor space and on the \(\text{SO}_3\)-principal bundle associated with the quaternionic structure. Another consequence is that a quaternionic Kähler solvmanifold admits quotients of finite volume if and only if it is a symmetric space. Finally, they give a simple description of the quaternionic Kähler solvmanifolds in terms of certain spinorial modules of \(\text{Spin}(3,3+k)\).