Principal Gelfand pairs

Let X = G/K be a connected Riemannian homogeneous space of a real Lie group G. The homogeneous space X is called commutative or the pair (G, K) is called a Gelfand pair if the algebra of G-invariant differential operators on X is commutative. We prove an effective commutativity criterion and classify Gelfand pairs under two mild technical constraints. In particular, we obtain several new examples of commutative homogeneous spaces that are not of Heisenberg type.