irreducible representation π of G the dimension of HomK(π,C) is less than or equal to 1. The representation L2(G/K) of G is multiplicity free i.e. it is a direct sum of distinct unitary irreducible representations. Lie group with compact subgroup When G is a Lie group and K is a compact subgroup the following are equivalent: (G,K) is a Gelfand pair. The algebra of (K,K)-double invariant compactly supported continuous measures on G with multiplication defined by convolution is commutative. The algebra D(G/K)K of K-invariant differential operators on G/K is commutative. For any continuous, locally convex, irreducible representation π of G, the space πK of K-invariant vectors in π is no-more-than-1-dimensional. For any continuous, locally convex, irreducible representation π of G the dimension of HomK(π, C) is less than or equal to 1. The representation L2(G/K) of G is multiplicity free i.e. it is a direct integral of distinct unitary irreducible representations. For a classification of such Gelfand pairs see. Classical examples of such Gelfand pairs are (G,K), where G is a reductive Lie group and K is a maximal compact subgroup. Locally compact topological group with compact subgroup When G is a locally compact topological group and K is a compact subgroup the following are equivalent: (G,K) is a Gelfand pair. The algebra of (K,K)-double invariant compactly supported continuous measures on G with multiplication defined by convolution is commutative. For any continuous locally convex irreducible representation π of G, the space πK of K-invariant vectors in π is no-more-than-1-dimensional. For any continuous, locally convex, irreducible representation π of G, the dimension of HomK(π, C) is less than or equal to 1. The representation L2(G/K) of G is multiplicity free i.e. it is a direct integral of distinct unitary irreducible representations. In that setting, G has an Iwasawa-Monod decomposition, namely G = K P for some amenable subgroup P of G. This is the abstract analogue of the Iwasawa decomposition of semisimple Lie groups. Lie group with closed subgroup When G is a Lie group and K is a closed subgroup, the pair (G,K) is called a generalized Gelfand pair if for any irreducible unitary representation π of G on a Hilbert space the dimension of HomK(π, C) is less than or equal to 1, where π∞ denotes the subrepresentation of smooth vectors. Reductive group over a local field with closed subgroup When G is a reductive group over a local