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 π of G, the space π^K of K-invariant vectors in π is no more than 1 dimensional.
• For any irreducible representation π of G, the dimension of Hom_K(π, 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 distinct absolutely irreducible representations in 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 compactly supported continuous measures on G with multiplication defined by convolution is commutative.
• For any continuous, locally convex, irreducible representation π of G, the space π^K of K-invariant vectors in π is no more than 1 dimensional.
• For any continuous, locally convex, irreducible representation π of G the dimension of Hom_K(π,C) is less than or equal to 1.
• The representation L²(G/K) of G is multiplicity free i.e. it is a direct sum of distinct 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 compactly supported continuous measures on G with multiplication defined by convolution is commutative.
• The algebra D(G/K)^K of K-invariant differential operators on G/K is commutative.
• For any continuous, locally convex, irreducible representation π of G, the space π^K of K-invariant vectors in π is no more than 1 dimensional.
• For any continuous, locally convex, irreducible representation π of G the dimension of Hom_K(π, C) is less than or equal to 1.
• The representation L²(G/K) of G is multiplicity free i.e. it is a direct integral of distinct unitary irreducible representations.

For a classification of such Gelfand pairs see ^[1].

Classical examples of such Gelfand pairs are (G,K), where G is a 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 compactly supported continuous measures on G with multiplication defined by convolution is commutative.
• For any continuous locally convex irreducible representation π of G, the space π^K of K-invariant vectors in π is no more than 1 dimensional.
• For any continuous, locally convex, irreducible representation π of G, the dimension of Hom_K(π, C) is less than or equal to 1.
• The representation L²(G/K) of G is multiplicity free i.e. it is a direct integral of distinct unitary irreducible representations.

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.

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 π of G on a Hilbert space the dimension of Hom_K(π, C) is less than or equal to 1, where π^∞ denotes the subrepresentation of 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 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 Hom_K(π, C) is less than or equal to 1.

GP2) For any irreducible admissible representation π of G we have ${\displaystyle \dim \mathrm {Hom} _{K}(\pi ,\mathbf {C} )\cdot \dim \mathrm {Hom} _{K}({\tilde {\pi }},\mathbf {C} )\leq 1}$, where ${\displaystyle {\tilde {\pi }}}$ denotes the smooth dual.

GP3) For any irreducible unitary representation π of G on a Hilbert space the dimension of Hom_K(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.

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 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 ^[3][4][5].

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 ^[6].

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.

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 called K-distinguished if Hom_K(π,C) is 1 dimensional. The representation Ind^G
_K(C) is a model for all K-distinguished representations i.e. 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, 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 twisted Gelfand pair w.r.t. any non-degenerate character ψ (see ^[3][13]). In this case K-distinguished representations are called generic (or non-degenerate) and they are easy to classify. Almost any irreducible representation is generic. The unique (up to scalar) imbedding of a generic representation to Ind^G
_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_i and characters ${\displaystyle \psi _{i}}$ s.t. (G,K_i) is twisted Gelfand pair w.r.t. ${\displaystyle \psi _{i}}$ and any irreducible unitary representation is K_i distinguished for exactly one i (see ^[14][15]).

Gelfand–Zeitlin construction

One can also use Gelfand pairs for constructing bases for irreducible representations: suppose we have a sequence {1} ⊂ G₁ ⊂ ... ⊂ G_n s.t. (G_i,G_i−1) is a strong Gelfand pair. For simplicity let's assume that G_n is compact. Then this gives a canonical decomposition of any irreducible representation of G_n to 1 dimensional subrepresentations. When G_n = 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 ${\displaystyle \nu }$ of F the pair (G, K) is a Gelfand pair over the completion ${\displaystyle F_{\nu }}$. 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 ${\displaystyle \nu }$).

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.

Now the Gelfand property for the pair (G,K) is an analog of the Schur's lemma.

Using this approach one can take any concepts of representation theory and generalize them to the case of spherical pair. For example, the relative trace formula is obtained from the trace formula by this procedure.

Finite groups

A few common examples of Gelfand pairs are:

• (Sym(n + 1), Sym(n)), the symmetric group acting on n + 1 points and a point stabilizer that is naturally isomorphic to on n points.
• (AGL(n, q), GL(n, q)), the 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 ${\displaystyle [1_{K},\chi \downarrow _{K}^{G}]\leq 1}$ for every χ in Irr(G). Since ${\displaystyle [1_{K},\chi \downarrow _{K}^{G}]=[1\uparrow _{K}^{G},\chi ]}$ by Frobenius reciprocity and ${\displaystyle 1\uparrow _{K}^{G}}$ 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 groups in (Breuer & Lux 1996).

This gives rise to a class of examples of finite groups with Gelfand pairs: the 2-transitive groups. A permutation group G is 2-transitive if the stabilizer K of a point acts 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 ≥ 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 1, (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, (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).

The Gelfand pairs (Sym(n),K) were classified in (Saxl 1981). Roughly speaking, K must be contained as a subgroup of small 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 n even, Sym(n − 5) × AGL(1,5), Sym(n − 6) × PGL(2,5), or 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 + k, R), O(n, R) × O(k, R))
• (U(n + 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 + 1, F), GL(n, F)) for n > 5.
• (Sp(2n + 2, F), Sp(2n, F)) × Sp(2, F)) for n > 4.
• (SO(V ⊕ F), SO(V)) where V is a vector space over F with a 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, ΔG): Follows from Schur's lemma.
• (GL(n + k, F), GL(n, F) × GL(k, F)).^[9][10]
• (GL(2n, F), Sp(2n, F)).^[16][17]
• (O(n + k, C), O(n, C) × O(k, C)).^[18]
• (GL(n, C), O(n, C)).^[18]
• (GL(n, E), GL(n, F)), where E is a quadratic extension of F.^[11][19]

Strong Gelfand pairs

The following pairs are strong Gelfand pairs:

• (Sym(n + 1), Sym(n)), this is proven using the involutive anti-automorphism g ↦ g⁻¹.
• (GL(n + 1, F), GL(n, F)) where F is a local field of characteristic zero.^[20][21][22]
• (O(V ⊕ F), O(V)) where V is a vector space over F with a non-degenerate quadratic form.^[20][22]
• U(V ⊕ E), U(V)) where E is a quadratic extension of F and V is a vector space over E with a non-degenerate hermitian form.^[20][22]

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

• (Sym(n + 1) × Sym(n), Δ Sym(n)).
• (GL(n + 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))