When G is a finite group, the simplest definition is, roughly speaking, that the (K, K )-double cosets in G commute.
More precisely, the Hecke algebra, the algebra of functions on G that are invariant under translation on either side by K, should be commutative for the convolution on G. In general, the definition of Gelfand pair is roughly that the restriction to K of any irreducible representation of G contains the trivial representation of K with multiplicity no more than 1.
[1] Classical examples of such Gelfand pairs are (G, K), where G is a reductive Lie group and K is a maximal compact subgroup.
When G is a locally compact topological group and K is a compact subgroup, the following are equivalent: In that setting, G has an Iwasawa–Monod decomposition, namely G = K P for some amenable subgroup P of G.[2] This is the abstract analogue of the Iwasawa decomposition of semisimple Lie groups.
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.
When G is a reductive group over a local field and K is a closed subgroup, there are three (possibly non-equivalent) notions of the Gelfand pair appearing in the literature:(GP1) For any irreducible admissible representation π of G, the dimension of HomK(π, C) is less than or equal to 1.
If the local field is Archimedean, then GP3 is the same as the generalized Gelfand property defined in the previous case.
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.
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.
If the above statement holds only for positive definite distributions, then the pair satisfies GP3 (see the next case).
Suppose that there exists an involutive anti-automorphism σ of G such that any K × K invariant positive definite distribution on G is σ-invariant.
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. 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 the case when K is compact, it means that the dimension of HomK(π, χ) is less than or equal to 1.
[citation needed] The Gelfand property is often satisfied by symmetric pairs.
If G is a connected reductive group over R and K = Gθ is a compact subgroup, then (G, K) is a Gelfand pair.
In general, it is an interesting question when a symmetric pair of a reductive group over a local field has the Gelfand property.
An irreducible representation of G is called K-distinguished if HomK(π, C) is one-dimensional.
In this case, K-distinguished representations are called generic (or non-degenerate) and are easy to classify.
The unique (up to scalar) imbedding of a generic representation to IndGK(ψ) is called a Whittaker model.
In the case of G = GL(n) there is a finer version of the result above; namely, there exist a finite sequence of subgroups Ki and characters
Then this gives a canonical decomposition of any irreducible representation of Gn to one-dimensional subrepresentations.
A more recent use of Gelfand pairs is for the 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
Then we will be led to the question of harmonic analysis on the space G/K with regard to the action of G. Now the Gelfand property for the pair (G, K) is an analog of the Schur's lemma.
Using this approach, any concept of representation theory can be generalized to the case of spherical pair.
The action of G on the cosets of K is thus faithful, so one is then looking at permutation groups G with point stabilizers K. To be a Gelfand pair is equivalent to
Such multiplicity-free permutation characters were determined for the sporadic groups in (Breuer & Lux 1996).
Indeed, if G is a transitive permutation group whose point stabilizer K has at most four orbits (including the trivial orbit containing only the stabilized point), then its Schur ring is commutative and (G, K) is a Gelfand pair, (Wielandt 1964, p. 86).
If G is a primitive group of degree twice a prime with point stabilizer K, then again (G, K) is a Gelfand pair, (Wielandt 1964, p. 97).