Gelfand pair w.r.t. and any irreducible unitary representation is Ki distinguished for exactly one i (see,) Gelfand–Zeitlin construction One can also use Gelfand pairs for constructing bases for irreducible representations: suppose we have a sequence {1} ⊂ G1 ⊂ ... ⊂ Gn s.t. (Gi,Gi-1) is a strong Gelfand pair. For simplicity let's assume that Gn is compact. Then this gives a canonical decomposition of any irreducible representation of Gn to one-dimensional subrepresentations. When Gn = U(n) (the unitary group) this construction is called Gelfand Zeitlin basis. Since the representations of U(n) are the same as algebraic representations of GL(n) so we also obtain a basis of any algebraic irreducible representation of GL(n). However one should be aware that the constructed basis isn't canonical as it depends on the choice of the embeddings U(i) ⊂ U(i+1). Splitting of periods of automorphic forms A more recent use of Gelfand pairs is for splitting of periods of automorphic forms. Let G be a reductive group defined over a global field F and let K be an algebraic subgroup of G. Suppose that for any place of F the pair (G, K) is a Gelfand pair over the completion . Let m be an automorphic form over G, then its H-period splits as a product of local factors (i.e. factors that depends only on the behavior of m at each place ). Now suppose we are given a family of automorphic forms with a complex parameter s. Then the period of those forms is an analytic function which splits into a product of local factors. Often this means that this function is a certain L-function and this gives an analytic continuation and functional equation for this L-function. Remark: usually those periods do not converge and one should regularize them. Generalization of representation theory A possible approach to representation theory is to consider representation theory of a group G as a harmonic analysis on the group G w.r.t. the two sided action of G × G. Indeed, to know all the irreducible representations of G is equivalent to know the decomposition of the space of functions on G as a G × G representation. In this approach representation theory can be generalized by replacing the pair (G × G, G) by any spherical pair (G,K). Then we will be led to the question of harmonic analysis on the space G/K w.r.t. the action of G.