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".
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 of G, the space K of K-invariant vectors in is no more than one-dimensional. For any irreducible representation of G, the dimension of HomK(, 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 of G, the space K of K-invariant vectors in is no more than one-dimensional. For any continuous, locally convex, irreducible representation of G, the dimension of HomK(,C) is less than or equal to 1. The representation L2(G/K) of G is multiplicity-free; that is, 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)G of G-invariant differential operators on G/K is commutative. For any continuous, locally convex, irreducible representation of G, the space K of K-invariant vectors in is no more than one-dimensional. For any continuous, locally convex, irreducible representation of G, the dimension of HomK(, C) is less than or equal to 1. The representation L2(G/K) of G is multiplicity-free; that is, it is a direct integral of distinct unitary irreducible representations.
For a classification of such Gelfand pairs, see.
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 of G, the space K of K-invariant vectors in is no more than one-dimensional. For any continuous, locally convex, irreducible representation of G, the dimension of HomK(, C) is less than or equal to 1. The representation L2(G/K) of G is multiplicity-free; that is, 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.Nicolas Monod, "Gelfand pairs admit an Iwasawa decomposition". 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 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.
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:() For any irreducible admissible representation of G, the dimension of HomK(, C) is less than or equal to 1.
() For any irreducible admissible representation of G, we have , where denotes the smooth dual.
() 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 is the same as the generalized Gelfand property defined in the previous case.
Clearly, ⇒ ⇒ .
Strong Gelfand pairs 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: . 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 of G and τ of K, the space HomK(τ,) is no more than one-dimensional. For any irreducible representations of G and τ of K, the space HomK(,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 Gelfand for the pair (G, K) to be Gelfand: Suppose that there exists an 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 Kazhdan for the pair (G, K) to satisfy . 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 (see 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).A. Aizenbud, D. Gourevitch, E. Sayag : (GL_{n+1}(F),GL_n(F)) is a Gelfand pair for any local field F. ).
If the above statement holds only for positive definite distributions, then the pair satisfies (see the next case).
The property often follows from . 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 involutive anti-automorphism σ of G such that any K × K invariant positive definite distribution on G is σ-invariant. Then the pair (G, K) is a generalized Gelfand pair (see 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).).
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 HomK(, χ) is less than or equal to 1. The criterion for Gelfand pairs can be adapted to the case of twisted Gelfand pairs.
Symmetric 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 investigations of symmetric pairs of rank one, see.
An example of high-rank Gelfand symmetric pair is . This was proven in Hervé Jacquet, Stephen Rallis (1996). Uniqueness of linear periods, Compositio Mathematica, tome 102, n.o. 1, pp. 65–123. 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.
Spherical pairs In the context of 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 of G, we have .
In this case, the space G/H is called spherical space.
It is conjectured that any spherical pair (G, K) over a local field satisfies the following weak version of the Gelfand property: For any admissible representation of G, the space HomK(, C) is finite-dimensional; moreover, the bound for this dimension does not depend on . This conjecture is proven for a large class of spherical pairs including all the symmetric pairs.Yiannis Sakellaridis and Akshay Venkatesh, "Periods and harmonic analysis on spherical varieties".
Applications
Classification Gelfand pairs are often used for classification of irreducible representations in the following way: