Gelfand pair

In mathematics, a Gelfand pair is a pair (G,K ) consisting of a group G and a subgroup K (called an Euler subgroup of G) that satisfies a certain property on restricted representations. The theory of Gelfand pairs is closely related to the topic of spherical functions in the classical theory of special functions, and to the theory of Riemannian symmetric spaces 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 ) 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. In each case, one should specify the class of considered representations and the meaning of "contains".

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

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 of G on a Hilbert space the dimension of HomK(, C) is less than or equal to 1, where ∞ 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. We will call them here GP1, GP2, and GP3. GP1) For any irreducible admissible representation of G the dimension of HomK(, C) is less than or equal to 1. GP2) For any irreducible admissible representation of G we have , where denotes the smooth dual. GP3) For any irreducible unitary representation of G on a Hilbert space the dimension of HomK(, 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 the generalized Gelfand property defined in the previous case. Clearly, GP1 ⇒ GP2 ⇒ GP3.

A pair (G,K) is called a strong Gelfand pair if the pair (G × K, ΔK) is a Gelfand pair, where ΔK ≤ G × K is the diagonal subgroup: {(k,k) in G × K : k in K}. 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 representation of G and τ of K, the space HomK(τ,) is no more than 1 dimensional. For any irreducible representation of G and τ of K, the space HomK(,τ) is no more than 1 dimensional.

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.

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.

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.

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 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 the case when K is compact it meanes that the dimension of HomK(, χ)) is less than or equal to 1. One can adapt the criterion for Gelfand pairs to the case of twisted Gelfand 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 investigated in and

An example of high rank Gelfand symmetric pair is (GL(n + k), GL(n) × L(k)). This was proven in over non archimedean local fields and later in for all local fields of characteristic zero. For more details on this question for high rank symmetric pairs see.

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 HomK(,C) is 1 dimensional. The representation Ind(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 ). 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(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 s. t. (G,Ki) is twisted Gelfand pair w. r. t. and any irreducible unitary representation is Ki distinguished for exactly one i (see ).

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 us assume that Gn is compact. Then this gives a canonical decomposition of any irreducible representation of Gn to 1 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 is not canonical as it depends on the choice of the embeddings U(i) ⊂ U(i+1).

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 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 depend 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 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. Remark: usually those periods do not converge and one should regularize them.

A possible approach to representation theory is to consider the 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.

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.

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)). (O(n + k, C), O(n, C) × O(k, C)). (GL(n, C), O(n, C)).