a local field with closed subgroup In this case there is a criterion due to Gelfand and Kazhdan for the pair (G,K) to satisfy GP2. Suppose that there exists an involutive anti-automorphism σ of G such that any (K,K)-double invariant distribution on G is σ-invariant. Then the pair (G,K) satisfies GP2. See. If the above statement holds only for positive definite distributions then the pair satisfies GP3 (see the next case). The property GP1 often follows from GP2. For example, this holds if there exists an involutive anti-automorphism of G that preserves K and preserves every closed conjugacy class. For G = GL(n) the transposition can serve as such involution. Lie group with closed subgroup In this case there is the following criterion for the pair (G,K) to be generalized Gelfand pair. Suppose that there exists an involutive anti-automorphism σ of G s.t. any K × K invariant positive definite distribution on G is σ-invariant. Then the pair (G,K) is a generalized Gelfand pair. See. Criteria for strong Gelfand property All the above criteria can be turned into criteria for strong Gelfand pairs by replacing the two-sided action of K × K by the conjugation action of K. Twisted Gelfand pairs A generalization of the notion of Gelfand pair is the notion of twisted Gelfand pair. Namely a pair (G,K) is called a twisted Gelfand pair with respect to the character χ of the group K, if the Gelfand property holds true when the trivial representation is replaced with the character χ. For example, in case when K is compact it meanes that the dimension of HomK(π, χ)) is less than or equal to 1. One can adapt the criterion for Gelfand pairs to the case of twisted Gelfand pairs. Symmetric pairs The Gelfand property is often satisfied by symmetric pairs. A pair (G,K) is called a symmetric pair if there exists an involutive automorphism θ of G such that K is a union of connected components of the group of θ-invariant elements: Gθ. If G is a connected reductive group over R and K=Gθ is a compact subgroup then (G,K) is a Gelfand pair. Example: G = GL(n,R) and K = O(n,R), the subgroup of orthogonal matrices. In general, it is an interesting question when a symmetric pair of a reductive group over a local field has the Gelfand property. For symmetric pairs of rank one this question was