field and K is a closed subgroup, there are three (possibly non-equivalent) notions of Gelfand pair appearing in the literature. We will call them here GP1, GP2, and GP3. GP1) For any irreducible admissible representation π of G the dimension of HomK(π, C) is less than or equal to 1. GP2) For any irreducible admissible representation π of G we have , where denotes the smooth dual. GP3) 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. Here, admissible representation is the usual notion of admissible representation when the local field is non-archimedean. When the local field is archimedean, admissible representation instead means smooth Fréchet representation of moderate growth such that the corresponding Harish-Chandra module is admissible. If the local field is archimedean, then GP3 is the same as generalized Gelfand property defined in the previous case. Clearly, GP1 ⇒ GP2 ⇒ GP3. Strong Gelfand pairs A pair (G,K) is called a strong Gelfand pair if the pair (G × K, ΔK) is a Gelfand pair, where ΔK ≤ G × K is the diagonal subgroup: {(k,k) in G × K : k in K}. Sometimes, this property is also called the multiplicity one property. In each of the above cases can be adapted to strong Gelfand pairs. For example, let G be a finite group. Then the following are equivalent. (G,K) is a strong Gelfand pair. The algebra of functions on G invariant with respect to conjugation by K (with multiplication defined by convolution) is commutative. For any irreducible representation π of G and τ of K, the space HomK(τ,π) is no-more-than-1-dimensional. For any irreducible representation π of G and τ of K, the space HomK(π,τ) is no-more-than-1-dimensional. Criteria for Gelfand property Locally compact topological group with compact subgroup In this case there is a classical criterion due to Gelfand for the pair (G,K) to be Gelfand: Suppose that there exists an involutive anti-automorphism σ of G s.t. any (K,K) double coset is σ invariant. Then the pair (G,K) is a Gelfand pair. This criterion is equivalent to the following one: Suppose that there exists an involutive anti-automorphism σ of G such that any function on G which is invariant with respect to both right and left translations by K is σ invariant. Then the pair (G,K) is a Gelfand pair. Reductive group over