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In [[mathematics]], a '''Gelfand pair''' is a pair (''G'', ''K''&hairsp;) consisting of a [[group (mathematics)|group]] ''G'' and a [[subgroup]] ''K'' (called a '''Euler subgroup''' of ''G'') that satisfies a certain property on [[restricted representation]]s. The theory of Gelfand pairs is closely related to the topic of [[Zonal spherical function|spherical function]]s in the classical theory of [[special functions]], and to the theory of [[Riemannian symmetric space]]s in [[differential geometry]]. Broadly speaking, the theory exists to abstract from these theories their content in terms of [[harmonic analysis]] and [[representation theory]].

When ''G'' is a [[finite group]], the simplest definition is, roughly speaking, that the (''K'', ''K''&hairsp;)-double [[coset]]s in ''G'' commute. More precisely, the [[Hecke algebra of a finite group|Hecke algebra]], the [[algebra over a field|algebra]] of [[function (mathematics)|function]]s on ''G'' that are invariant under translation on either side by ''K'', should be [[commutative algebra (structure)|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. In each case, one should specify the class of considered representations and the meaning of "contains".

==Definitions==
In each area, the class of representations and the definition of containment for representations is slightly different. Explicit definitions of several such cases are given here.

===Finite group case===
When ''G'' is a finite group, the following are equivalent:

* (''G'', ''K'') is a Gelfand pair.
* The algebra of (''K'', ''K'')-double invariant functions on ''G'' with multiplication defined by convolution is commutative.
* For any irreducible representation {{pi}} of ''G'', the space {{pi}}<sup>''K''</sup> of ''K''-[[invariant (mathematics)|invariant]] vectors in {{pi}} is no more than one-dimensional.
* For any irreducible representation {{pi}} of ''G'', the dimension of Hom<sub>''K''</sub>({{pi}}, '''C''') is less than or equal to 1, where '''C''' denotes the [[trivial representation]].
* The [[permutation representation]] of ''G'' on the cosets of ''K'' is multiplicity-free; that is, it decomposes into a [[direct sum of representations|direct sum]] of distinct [[absolutely irreducible]] representations in [[characteristic (algebra)|characteristic]] zero.
* The [[centralizer algebra]] ([[Schur algebra]]) of the permutation representation is commutative.
* (''G''/''N'', ''K''/''N'') is a Gelfand pair, where ''N'' is a [[normal subgroup]] of ''G'' contained in ''K''.

===Compact group case===
When ''G'' is a [[compact topological group]], the following are equivalent:

* (''G'', ''K'') is a Gelfand pair.
* The algebra of (''K'', ''K'')-double invariant [[Support (mathematics)#Compact support|compactly supported]] continuous [[measure (mathematics)|measure]]s on ''G'' with multiplication defined by convolution is commutative.
* For any [[Group action#Continuous group actions|continuous]], [[Locally convex topological vector space|locally convex]], irreducible representation {{pi}} of ''G'', the space {{pi}}<sup>''K''</sup> of ''K''-[[invariant (mathematics)|invariant]] vectors in {{pi}} is no more than one-dimensional.
* For any continuous, locally convex, irreducible representation {{pi}} of ''G'', the dimension of Hom<sub>''K''</sub>({{pi}},'''C''') is less than or equal to 1.
* The representation ''L''<sup>2</sup>(''G''/''K'') of ''G'' is multiplicity-free; that is, it is a direct sum of distinct [[unitary representation|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 [[Support (mathematics)#Compact support|compactly supported]] continuous [[measure (mathematics)|measure]]s on ''G'' with multiplication defined by convolution is commutative.
* The algebra ''D''(''G''/''K'')<sup>''G''</sup> of ''G''-invariant [[differential operator]]s on ''G''/''K'' is commutative.
* For any [[Group action#Continuous group actions|continuous]], [[Locally convex topological vector space|locally convex]], irreducible representation {{pi}} of ''G'', the space {{pi}}<sup>''K''</sup> of ''K''-[[invariant (mathematics)|invariant]] vectors in {{pi}} is no more than one-dimensional.
* For any continuous, locally convex, irreducible representation {{pi}} of ''G'', the dimension of Hom<sub>''K''</sub>({{pi}}, '''C''') is less than or equal to 1.
* The representation ''L''<sup>2</sup>(''G''/''K'') of ''G'' is multiplicity-free; that is, it is a [[direct integral]] of distinct [[unitary representation|unitary]] irreducible representations.

For a classification of such Gelfand pairs, see.<ref name="Yak">{{Cite thesis |title=Gelfand pairs |url=https://nbn-resolving.org/urn:nbn:de:hbz:5N-05134 |institution=Rheinische Friedrich-Wilhelms-Universität Bonn |date=2005 |first=Oksana |last=Yakimova}}</ref>

Classical examples of such Gelfand pairs are (''G'', ''K''), where ''G'' is a [[Reductive group#Lie group case|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 [[Support (mathematics)#Compact support|compactly supported]] continuous [[measure (mathematics)|measure]]s on ''G'' with multiplication defined by convolution is commutative.
* For any [[Group action#Continuous group actions|continuous]] [[locally convex topological vector space|locally convex]] [[irreducible representation]] {{pi}} of ''G'', the space {{pi}}<sup>''K''</sup> of ''K''-[[invariant (mathematics)|invariant]] vectors in {{pi}} is no more than one-dimensional.
* For any continuous, locally convex, irreducible representation {{pi}} of ''G'', the dimension of Hom<sub>''K''</sub>({{pi}}, '''C''') is less than or equal to 1.
* The representation ''L''<sup>2</sup>(''G''/''K'') of ''G'' is multiplicity-free; that is, it is a [[direct integral]] of distinct [[unitary representation|unitary]] irreducible representations.

In that setting, ''G'' has an [[Kenkichi Iwasawa|Iwasawa]]–[[Nicolas Monod|Monod]] decomposition, namely ''G = K P'' for some [[amenable group|amenable]] subgroup ''P'' of ''G''.<ref name=GKP>[[Nicolas Monod]], "Gelfand pairs admit an Iwasawa decomposition". {{arxiv|1902.09497}}</ref> This is the abstract analogue of the [[Iwasawa decomposition]] of [[semisimple Lie group]]s.

===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]] {{pi}} of'' G'' on a [[Hilbert space]], the dimension of Hom<sub>''K''</sub>({{pi}}, '''C''') is less than or equal to 1, where {{pi}}<sup>∞</sup> denotes the subrepresentation of [[Group action#Continuous group actions|smooth vectors]].

===Reductive group over a local field with closed subgroup===
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:<blockquote>({{EquationRef|GP1}}) For any irreducible admissible representation {{pi}} of ''G'', the dimension of Hom<sub>''K''</sub>({{pi}}, '''C''') is less than or equal to 1.

({{EquationRef|GP2}}) For any irreducible admissible representation {{pi}} of ''G'', we have <math display="inline">\dim \operatorname{Hom}_K(\pi, \mathbf{C}) \cdot \dim \operatorname{Hom}_K(\tilde{\pi}, \mathbf{C}) \leq 1</math>, where <math>\tilde{\pi}</math> denotes the [[Group action#Continuous group actions|smooth]] [[dual representation|dual]].

({{EquationRef|GP3}}) For any irreducible [[unitary representation]] {{pi}} of ''G'' on a [[Hilbert space]], the dimension of Hom<sub>''K''</sub>({{pi}}, '''C''') is less than or equal to 1.</blockquote>Here, ''admissible representation'' is the usual notion of [[admissible representation]] when the local field is non-[[Archimedean property|Archimedean]]. When the local field is Archimedean, ''admissible representation'' instead means smooth [[Fréchet space|Fréchet]] [[representation of moderate growth]] such that the corresponding Harish–Chandra module is [[admissible representation#Real reductive groups|admissible]].

If the local field is Archimedean, then {{EquationNote|GP3}} is the same as the generalized Gelfand property defined in the previous case.

Clearly, {{EquationNote|GP1}} ⇒ {{EquationNote|GP2}} ⇒ {{EquationNote|GP3}}.

==Strong Gelfand pairs==
A pair (''G'', ''K'') is called a '''strong Gelfand pair''' if the pair (''G'' × ''K'', Δ''K'') is a Gelfand pair, where Δ''K'' ≤ ''G'' × ''K'' is the diagonal subgroup: <math display="inline">\{(k,k) \in G \times K:k \in K\}</math>. Sometimes, this property is also called the '''multiplicity one''' property.

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 representations]] {{pi}} of ''G'' and ''τ'' of ''K'', the space Hom<sub>''K''</sub>(''τ'',{{pi}}) is no more than one-dimensional.
* For any irreducible representations {{pi}} of ''G'' and ''τ'' of ''K'', the space Hom<sub>''K''</sub>({{pi}},''τ'') is no more than one-dimensional.

==Criteria for Gelfand property==

===Locally compact topological group with compact subgroup===
In this case, there is a classical criterion due to [[Israel Gelfand|Gelfand]] for the pair (''G'', ''K'') to be Gelfand: Suppose that there exists an [[involution (mathematics)|involutive]] [[anti-automorphism]] ''σ'' of ''G'' such that 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 a local field with closed subgroup===
In this case, there is a criterion due to Gelfand and [[David Kazhdan|Kazhdan]] for the pair (''G'', ''K'') to satisfy {{EquationNote|GP2}}. Suppose that there exists an [[involution (mathematics)|involutive]] [[anti-automorphism]] ''σ'' of ''G'' such that any (''K'', ''K'')-double invariant [[distribution (mathematics)|distribution]] on ''G'' is ''σ''-invariant. Then the pair (''G'', ''K'') satisfies {{EquationNote|GP2}} (see <ref name=GK>[[Israel Gelfand]], [[David Kazhdan]], Representations of the group GL(n,K) where K is a local field, Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971), pp. 95--118. Halsted, New York (1975).</ref><ref name=AGS>A. Aizenbud, D. Gourevitch, E. Sayag : (GL_{n+1}(F),GL_n(F)) is a Gelfand pair for any local field F. {{arxiv|0709.1273}}</ref><ref name=SZ11>{{Citation | last1=Sun | first1=Binyong |authorlink1 = Sun Binyong | last2=Zhu |first2=Chen-Bo | authorlink2 = Chen-Bo Zhu | title=A general form of Gelfand-Kazhdan criterion | year=2011 | journal=Manuscripta Math.| mr=2820401 | volume=136 | issue=1–2 | pages=185–197| doi=10.1007/s00229-011-0437-x | arxiv=0903.1409 }}</ref>).

If the above statement holds only for [[positive definite function on a group|positive definite]] distributions, then the pair satisfies {{EquationNote|GP3}} (see the next case).

The property {{EquationNote|GP1}} often follows from {{EquationNote|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 an involution.

===Lie group with closed subgroup===
In this case, there is the following criterion for the pair (''G'', ''K'') to be a generalized Gelfand pair. Suppose that there exists an [[involution (mathematics)|involutive]] [[anti-automorphism]] ''σ'' of ''G'' such that any ''K'' × ''K'' invariant positive definite [[distribution (mathematics)|distribution]] on ''G'' is ''σ''-invariant. Then the pair (''G'', ''K'') is a generalized Gelfand pair (see <ref name=Tho>E.G.F. Thomas, The theorem of Bochner-Schwartz-Godement for generalized Gelfand pairs, Functional Analysis: Surveys and results III, Bierstedt, K.D., Fuchssteiner, B. (eds.), Elsevier Science Publishers B.V. (North Holland), (1984).</ref>).

===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 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 Hom<sub>K</sub>({{pi}}, χ) is less than or equal to 1. The criterion for Gelfand pairs can be adapted to the case of twisted Gelfand pairs.{{Citation needed|date=July 2024}}

==Symmetric pairs==
The Gelfand property is often satisfied by ''symmetric pairs''. A pair (''G'', ''K'') is called a '''symmetric pair''' if there exists an [[involution (mathematics)|involutive]] [[automorphism]] ''θ'' of ''G'' such that ''K'' is a union of connected components of the group of ''θ''-invariant elements: ''G''<sup>''θ''</sup>.

If ''G'' is a [[connected space|connected]] [[reductive group]] over '''R''' and ''K''&nbsp;=&nbsp;''G''<sup>''θ''</sup> is a compact subgroup, then (''G'', ''K'') is a Gelfand pair. Example: ''G''&nbsp;=&nbsp;GL(''n'', '''R''') and ''K''&nbsp;=&nbsp;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 investigations of symmetric pairs of rank one, see.<ref name=vD>{{cite journal |first=Gerrit |last=van Dijk |title=On a class of generalized Gelfand pairs |journal=Math. Z. 193 |pages=581–593 |year=1986}}</ref><ref name=BvD>{{cite journal | last1 = Bosman | first1 = E. P. H. | last2 = Van Dijk | first2 = G. | year = 1994 | title = A New Class of Gelfand Pairs | journal = Geometriae Dedicata | volume = 50 | issue = 3| pages = 261–282 | doi=10.1007/bf01267869| s2cid = 121913299 }}</ref>

An example of high-rank Gelfand symmetric pair is <math display="inline">(\text{GL}(n + k), \text{GL}(n) \times \text{L}(k))</math>. This was [[mathematical proof|proven]] in <ref name=JR>[[Hervé Jacquet]], [[Stephen Rallis]] (1996). [http://archive.numdam.org/ARCHIVE/CM/CM_1996__102_1/CM_1996__102_1_65_0/CM_1996__102_1_65_0.pdf Uniqueness of linear periods], Compositio Mathematica, tome 102, n.o. 1 , pp. 65–123.</ref> over non-Archimedean local fields and later in <ref name=AG2>{{cite journal |first1=A. |last1=Aizenbud |first2=D. |last2=Gourevitch |title=(GL<sub> ''n'' +1</sub>( ''F'' ), GL<sub> ''n'' </sub>( ''F'' )) is a Gelfand pair for any local field ''F'' |journal=Compositio Mathematica |year=2007 |volume=144 |issue=6 |pages=1504–1524 |doi=10.1112/S0010437X08003746 |arxiv=0709.1273}}</ref> for all local fields of [[characteristic (algebra)|characteristic]] zero.

For more details on this question for high-rank symmetric pairs, see.<ref name=AG1>{{cite arXiv |first1=A. |last1=Aizenbud |first2=D. |last2=Gourevitch |title=Generalized Harish-Chandra descent and applications to Gelfand pairs |date=2008 |class=math.RT |eprint=0803.3395 }}</ref>

==Spherical pairs==
In the context of [[Algebraic group|algebraic groups]], the analogs of Gelfand pairs are called ''spherical pairs''. Namely, a pair (''G'', ''K'') of algebraic groups is called a '''spherical pair''' if one of the following equivalent conditions holds:

* There exists an open (''B'', ''K'')-double coset in ''G'', where ''B'' is the [[Borel subgroup]] of ''G''.
* There is a finite number of (''B'', ''K'')-double coset in ''G''.
* For any algebraic representation {{pi}} of ''G'', we have <math>\text{dim} \ \pi^K \leq 1</math>.

In this case, the space ''G''/''H'' is called ''spherical space''.

It is [[conjecture]]d that any spherical pair (''G'', ''K'') over a local field satisfies the following weak version of the Gelfand property:
For any admissible representation {{pi}} of ''G'', the space Hom<sub>''K''</sub>({{pi}}, '''C''') is finite-dimensional; moreover, the bound for this dimension does not depend on {{pi}}. This conjecture is proven for a large class of spherical pairs including all the symmetric pairs.<ref name=SV>Yiannis Sakellaridis and [[Akshay Venkatesh]], "Periods and harmonic analysis on spherical varieties". {{arxiv|1203.0039}}</ref>

==Applications==

===Classification===
Gelfand pairs are often used for classification of irreducible representations in the following way:

Let (''G'', ''K'') be a Gelfand pair. An irreducible representation of G is called ''K''-distinguished if Hom<sub>''K''</sub>({{pi}}, '''C''') is one-dimensional. The representation Ind{{su|p=G|b=K}}('''C''') is a model for all ''K''-distinguished representations, that is, any ''K''-distinguished representation appears there with multiplicity exactly 1. A similar notion exists for twisted Gelfand pairs.

'''Example:''' If ''G'' is a reductive group over a local field and K is its maximal compact subgroup, then ''K''-distinguished representations are called [[spherical representation|spherical]], and such representations can be classified via the [[Satake correspondence]]. The notion of spherical representation is in the basis of the notion of [[Harish-Chandra module]].

'''Example:''' If ''G'' is [[split reductive group]] over a local field and ''K'' is its [[maximal unipotent subgroup]], then the pair (''G'', ''K'') is a twisted Gelfand pair with regard to any [[non-degenerate character]] ''ψ'' (see <ref name="GK" /><ref name="Sha">[[Joseph Shalika]], The multiplicity one theorem for GL<sub>''n''</sub>, Ann. of Math. 100(1974) 171–193. {{MR|348047 }}</ref>). In this case, ''K''-distinguished representations are called [[generic representation|generic]] (or non-degenerate) and are easy to classify. Almost any irreducible representation is generic. The unique (up to scalar) imbedding of a generic representation to Ind{{su|p=G|b=K}}(''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 ''K<sub>i</sub>'' and characters <math>\psi_i</math>  such that (''G'', ''K<sub>i</sub>'') is a twisted Gelfand pair with regard to <math>\psi_i</math> and any irreducible unitary representation is ''K<sub>i</sub>'' distinguished for exactly one ''i'' (see <ref name="OS1">Omer Offen, Eitan Sayag, Global Mixed Periods and local Klyachko models for the general linear group, {{arxiv|0710.3492}}</ref><ref name="OS2">Omer Offen, Eitan Sayag, UNIQUENESS AND DISJOINTNESS OF KLYACHKO MODELS , {{arxiv|0710.3492}}</ref>).

===Gelfand–Zeitlin construction===
{{see also|Gelfand–Zeitlin integrable system}}
One can also use Gelfand pairs for constructing bases for irreducible representations.

Suppose we have a sequence <math display="inline">\{1\} \subset G_1 \subset \cdots \subset G_n</math> such that <math display="inline">(G_i, G_{i-1})</math> is a strong Gelfand pair. For simplicity let us assume that ''G<sub>n</sub>'' is compact. Then this gives a canonical decomposition of any irreducible representation of ''G<sub>n</sub>'' to one-dimensional subrepresentations. When ''G<sub>n</sub>'' = U(''n'') (the unitary group), this construction is called a [[Gelfand–Zeitlin basis]]. Since the representations of U(''n'') are the same as algebraic representations of GL(''n''), we also obtain a basis of any algebraic irreducible representation of GL(''n''). However, the constructed basis is not canonical as it depends on the choice of the embeddings <math display="inline">U(i) \subset U(i+1)</math>.

===Splitting of periods of automorphic forms===
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 of a global field|place]] <math>\nu</math> of ''F'', the pair (''G'', ''K'') is a Gelfand pair over the [[Completion (ring theory)|completion]] <math>F_\nu</math>. Let ''m'' be an [[automorphic form]] over ''G'', then its ''H''-period splits as a product of local factors (i.e. factors that depend only on the behavior of ''m'' at each place <math>\nu</math>).

Now suppose we are given a family of automorphic forms with a complex parameter&nbsp;''s''. Then the period of those forms is an analytic function that 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.

Usually those periods do not converge and one should regularize them.{{Citation needed|date=July 2024}}

===Generalization of representation theory===
A possible approach to representation theory is to consider the representation theory of a group ''G'' as a [[harmonic analysis]] on the group ''G'' with regard to 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'' 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. For example, the [[relative trace formula]] is obtained from the [[Arthur–Selberg trace formula|trace formula]] by this procedure.

==Examples==

===Finite groups===
A few common examples of Gelfand pairs are:
* <math>(\text{Sym}(n+1), \text{Sym}(n))</math>, the [[symmetric group]] acting on ''n''+1 points and a point stabilizer that is naturally isomorphic to on ''n'' points.
* <math>(\text{AGL}(n,q), \text{GL}(n,q))</math>, the [[affine group|affine (general linear) group]] and a point stabilizer that is naturally isomorphic to the [[general linear group]].

If (''G'', ''K'') is a Gelfand pair, then (''G''/''N'', ''K''/''N'') is a Gelfand pair for every ''G''-[[normal subgroup]] ''N'' of ''K''. For many purposes it suffices to consider ''K'' without any such non-identity normal subgroups. 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 <math> [1_K,\chi\downarrow^G_K] \leq 1</math> for every ''χ'' in Irr(''G''). Since <math>[1_K,\chi\downarrow^G_K] = [1\uparrow_K^G,\chi]</math> by [[Frobenius reciprocity]] and <math>1\uparrow_K^G</math> is the character of the permutation action, a permutation group defines a Gelfand pair if and only if the permutation character is a so-called '''multiplicity-free''' permutation character. Such multiplicity-free permutation characters were determined for the [[sporadic group]]s in {{harv|Breuer|Lux|1996}}.

This gives rise to a class of examples of finite groups with Gelfand pairs: the [[2-transitive group]]s. A [[permutation group]] ''G'' is '''2-transitive''' if the [[Group action#Orbits and stabilizers|stabilizer]] ''K'' of a point acts [[Group_action#Transitivity_properties|transitively]] on the remaining points. In particular, ''G'' the symmetric group on ''n''+1 points and ''K'' the symmetric group on ''n'' points forms a Gelfand pair for every ''n&nbsp;''≥&nbsp;1. This follows because the character of a 2-transitive permutation action is of the form 1 + ''χ'' for some irreducible character ''χ'' and the [[trivial character]]&nbsp;1, {{harv|Isaacs|1994|p=69}}.

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, {{harv|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, {{harv|Wielandt|1964|p=97}}.

The Gelfand pairs (Sym(''n''), ''K'') were classified in {{harv|Saxl|1981}}. Roughly speaking, ''K'' must be contained as a subgroup of small [[Index of a subgroup|index]] in one of the following groups unless ''n'' is smaller than 18:

* Sym(''n'' − ''k'') × Sym(''k'')
* Sym(''n''/2) wr Sym(2), Sym(2) wr Sym(''n''/2) for [[parity (mathematics)|even]] ''n'', where wr denotes the [[wreath product]]
* Sym(''n'' − 5) × AGL(1, 5)
* Sym(''n'' − 6) × PGL(2, 5)
* Sym(''n'' − 9) × PΓL(2, 8)

Gelfand pairs for classical groups have been investigated as well.

===Symmetric pairs with compact ''K''===
* (GL(''n'', '''R'''), O(''n'', '''R'''))
* (GL(''n'', '''C'''), U(''n''))
* (O(''n''&nbsp;+&nbsp;''k'', '''R'''), O(''n'', '''R''') × O(''k'', '''R'''))
* (U(''n''&nbsp;+&nbsp;''k''), U(''n'') × U(''k''))
* (''G'', ''K'') where ''G'' is a [[reductive Lie group]] and ''K'' is a [[maximal compact subgroup]]

===Symmetric Gelfand pairs of rank one===
Let ''F'' be a local field of characteristic zero.

* (SL(''n''&nbsp;+&nbsp;1, ''F''), GL(''n'', ''F'')) for ''n'' > 5
* (Sp(2''n''&nbsp;+&nbsp;2, ''F''), Sp(2''n'', ''F'')) × Sp(2, ''F'')) for ''n'' > 4
* (SO(''V'' ⊕ ''F''), SO(''V'')) where ''V'' is a vector space over ''F'' with a [[degenerate form|non-degenerate]] [[quadratic form]]

===Symmetric pairs of high rank===
Let ''F'' be a local field of characteristic zero. Let ''G'' be a [[reductive group]] over ''F''. The following are examples of symmetric Gelfand pairs of high rank:

* (''G'' × ''G'', Δ''G''), follows from [[Schur's lemma]]
* (GL(''n''&nbsp;+&nbsp;''k'', ''F''), GL(''n'', ''F'') × GL(''k'', ''F''))<ref name=JR/><ref name=AG2/>
* (GL(2''n'', ''F''), Sp(2''n'', ''F''))<ref name=HR>{{cite journal | last1 = Heumos | first1 = Michael J. | last2 = Rallis | first2 = Stephen |authorlink2 = Stephen Rallis | year = 1990 | title = Symplectic-Whittaker models for GLn | journal = Pacific J. Math. | volume = 146 | issue = 2| pages = 247–279 | doi=10.2140/pjm.1990.146.247| doi-access = free }}</ref><ref name=Say>E.Sayag (GL(2n,C),SP(2n,C)) is a Gelfand Pair {{arxiv|0805.2625}}</ref>
* (O(''n''&nbsp;+&nbsp;''k'', '''C'''), O(''n'', '''C''') × O(''k'', '''C'''))<ref name=AG3>A. Aizenbud, D. Gourevitch. Some regular symmetric pairs. {{arxiv|0805.2504}}</ref>
* (GL(''n'', '''C'''), O(''n'', '''C'''))<ref name=AG3/>
* (GL(''n'', ''E''), GL(''n'', ''F'')) where ''E'' is a [[quadratic extension]] of ''F''<ref name=AG1/><ref name=Fli>Y.Z. Flicker: On distinguished representations, J. Reine Angew. Math. 418 (1991), 139-172.</ref>

===Strong Gelfand pairs===
The following pairs are strong Gelfand pairs:

* (Sym(''n''&nbsp;+&nbsp;1), Sym(''n'')), proven using the involutive anti-automorphism ''g'' ↦ ''g''<sup>−1</sup>
* (GL(''n''&nbsp;+&nbsp;1, ''F''), GL(''n'', ''F'')) where ''F'' is a local field of characteristic zero<ref name=AGRS>{{Citation | last1=Aizenbud| first1=Avraham | last2=Gourevitch | first2=Dmitry | last3=Rallis | first3=Stephen |authorlink3 = Stephen Rallis | last4= Schiffmann | first4=Gérard | title=Multiplicity-one theorems | year=2010 | journal=[[Annals of Mathematics]] | volume=172 | issue=2 | pages=1407–1434 | mr=2680495 | doi=10.4007/annals.2010.172.1413 | arxiv=0709.4215 }}</ref><ref name=AG4>{{Citation | last1=Aizenbud| first1=Avraham | last2=Gourevitch | first2=Dmitry | title=Multiplicity one theorem for (GL(''n''&nbsp;+&nbsp;1,&nbsp;'''R'''),&nbsp;GL(''n'',&nbsp;'''R''')) | year=2009 | journal=Selecta Math. |series=New Series | mr=2529937 | volume=15 | issue=2 | pages=271–294| doi=10.1007/s00029-009-0544-7 | arxiv=0808.2729 }}</ref><ref name=SZ>{{Citation | last1=Sun | first1=Binyong |authorlink1 = Sun Binyong | last2=Zhu |first2=Chen-Bo |authorlink2 = Chen-Bo Zhu | title=Multiplicity-one theorems: the Archimedean case | year=2012 | journal=Annals of Mathematics | mr=2874638 | volume=175 | issue=1 | pages=23–44| doi=10.4007/annals.2012.175.1.2 | arxiv=0903.1413 }}</ref>
* (O(''V'' ⊕ ''F''), O(''V'')) where ''V'' is a vector space over ''F'' with a [[degenerate form|non-degenerate]] [[quadratic form]]<ref name=AGRS/><ref name=SZ/>
* U(''V'' ⊕ ''E''), U(''V'')) where ''E'' is a [[quadratic extension]] of ''F'' and ''V'' is a vector space over ''E'' with a [[degenerate form|non-degenerate]] [[Sesquilinear form#Hermitian form|Hermitian form]]<ref name=AGRS/><ref name=SZ/>

Those four examples can be rephrased as the statement that the following are Gelfand pairs:

* (Sym(''n''&nbsp;+&nbsp;1) × Sym(''n''), Δ Sym(''n''))
* (GL(''n''&nbsp;+&nbsp;1, ''F'') × GL(''n'', ''F''), Δ GL(''n'', ''F''))
* (O(''V'' ⊕ ''F'') × O(''V''), Δ O(''V''))
* (U(''V'' ⊕ ''E'') × U(''V''), Δ U(''V''))

==See also==
* [[Zonal spherical function|Spherical function]]

==References==
<references/>

==Works cited==
* {{Citation | last1=Breuer | first1=T. | last2=Lux | first2=K. | title=The multiplicity-free permutation characters of the sporadic simple groups and their automorphism groups | doi=10.1080/00927879608825701 |mr=1390375 | year=1996 | journal=Communications in Algebra | volume=24 | issue=7 | pages=2293–2316}}
* {{Citation | last1=Isaacs | first1=I. Martin | title=Character Theory of Finite Groups | publisher=[[Dover Publications]] | location=New York | isbn=978-0-486-68014-9 |mr=460423 | year=1994}}
* {{Citation | last1=Saxl | first1=Jan |author1-link=Jan Saxl| title=Finite geometries and designs (Proc. Conf., Chelwood Gate, 1980) | publisher=[[Cambridge University Press]] | series=London Math. Soc. Lecture Note Ser. |mr=627512 | year=1981 | volume=49 | chapter=On multiplicity-free permutation representations | pages=337–353}}
* {{citation|title=Introduction to Harmonic Analysis and Generalized Gelfand Pairs|volume= 36 |series= De Gruyter studies in mathematics|
first=Gerrit|last= van Dijk|publisher=Walter de Gruyter|year= 2009|isbn=978-3-11-022019-3}}
* {{Citation | last1=Wielandt | first1=Helmut | title=Finite permutation groups | publisher=[[Academic Press]] | location=Boston, MA |mr=0183775 | year=1964}}

{{DEFAULTSORT:Gelfand Pair}}
[[Category:Representation theory of groups]]
[[Category:Representation theory of Lie groups]]
[[Category:Harmonic analysis]]