In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a Gelfand pair is a pair ''(G,K)'' consisting of a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic iden ...
''G'' and a
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
''K'' (called an Euler subgroup of ''G'') that satisfies a certain property on
restricted representation In group theory, restriction forms a representation of a subgroup using a known representation of the whole group. Restriction is a fundamental construction in representation theory of groups. Often the restricted representation is simpler to under ...
s. The theory of Gelfand pairs is closely related to the topic of
spherical functions in the classical theory of
special function
Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications.
The term is defined by ...
s, and to the theory of
Riemannian symmetric space
In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, ...
s in
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
. Broadly speaking, the theory exists to abstract from these theories their content in terms of
harmonic analysis
Harmonic analysis is a branch of mathematics concerned with the representation of Function (mathematics), functions or signals as the Superposition principle, superposition of basic waves, and the study of and generalization of the notions of Fo ...
and
representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
.
When ''G'' is a
finite group
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked ...
the simplest definition is, roughly speaking, that the ''(K,K)''-double
coset
In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
s in ''G'' commute. More precisely, the
Hecke algebra
In mathematics, the Hecke algebra is the algebra generated by Hecke operators.
Properties
The algebra is a commutative ring.
In the classical elliptic modular form theory, the Hecke operators ''T'n'' with ''n'' coprime to the level acting on ...
, the algebra of functions on ''G'' that are invariant under translation on either side by ''K'', should be commutative for the
convolution
In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
on ''G''.
In general, the definition of Gelfand pair is roughly that the restriction to ''K'' of any
irreducible representation
In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W,W ...
of ''G'' contains the
trivial representation In the mathematical field of representation theory, a trivial representation is a representation of a group ''G'' on which all elements of ''G'' act as the identity mapping of ''V''. A trivial representation of an associative or Lie algebra is a ...
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 in several such cases are given here.
Finite group case
When ''G'' is a
finite group
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked ...
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
Invariant and invariance may refer to:
Computer science
* Invariant (computer science), an expression whose value doesn't change during program execution
** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...
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 In the mathematical field of representation theory, a trivial representation is a representation of a group ''G'' on which all elements of ''G'' act as the identity mapping of ''V''. A trivial representation of an associative or Lie algebra is a ...
.
* The
permutation representation
In mathematics, the term permutation representation of a (typically finite) group G can refer to either of two closely related notions: a representation of G as a group of permutations, or as a group of permutation matrices. The term also refers t ...
of ''G'' on the cosets of ''K'' is multiplicity-free, that is, it decomposes into a
direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
of distinct
absolutely irreducible In mathematics, a multivariate polynomial defined over the rational numbers is absolutely irreducible if it is irreducible over the complex field.. For example, x^2+y^2-1 is absolutely irreducible, but while x^2+y^2 is irreducible over the integers ...
representations in
characteristic zero.
* The
centralizer algebra
In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', ...
(
Schur algebra In mathematics, Schur algebras, named after Issai Schur, are certain finite-dimensional algebras closely associated with Schur–Weyl duality between general linear and symmetric groups. They are used to relate the representation theories of ...
) of the permutation representation is
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
.
* (''G''/''N'', ''K''/''N'') is a Gelfand pair, where ''N'' is a
normal subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G i ...
of ''G'' contained in ''K''.
Compact group case
When ''G'' is a
compact topological group
In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural gen ...
the following are equivalent:
* ''(G,K)'' is a Gelfand pair.
* The algebra of ''(K,K)''-double invariant
compactly supported
In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest ...
continuous
measures
Measure may refer to:
* Measurement, the assignment of a number to a characteristic of an object or event
Law
* Ballot measure, proposed legislation in the United States
* Church of England Measure, legislation of the Church of England
* Meas ...
on ''G'' with multiplication defined by convolution is commutative.
* For any
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
,
locally convex
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ve ...
, irreducible representation ''π'' of ''G'', the space ''π''
''K'' of ''K''-
invariant
Invariant and invariance may refer to:
Computer science
* Invariant (computer science), an expression whose value doesn't change during program execution
** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...
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 ''L''
2''(G/K)'' of ''G'' is multiplicity free i.e. it is a direct sum of distinct
unitary
Unitary may refer to:
Mathematics
* Unitary divisor
* Unitary element
* Unitary group
* Unitary matrix
* Unitary morphism
* Unitary operator
* Unitary transformation
* Unitary representation
* Unitarity (physics)
* ''E''-unitary inverse semigroup ...
irreducible representations.
Lie group with compact subgroup
When ''G'' is a
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
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
In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest ...
continuous
measures
Measure may refer to:
* Measurement, the assignment of a number to a characteristic of an object or event
Law
* Ballot measure, proposed legislation in the United States
* Church of England Measure, legislation of the Church of England
* Meas ...
on ''G'' with multiplication defined by convolution is commutative.
* The algebra ''D(G/K)''
''K'' of ''K''-invariant
differential operators
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...
on ''G/K'' is commutative.
* For any
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
,
locally convex
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ve ...
, irreducible representation ''π'' of ''G'', the space ''π''
''K'' of ''K''-
invariant
Invariant and invariance may refer to:
Computer science
* Invariant (computer science), an expression whose value doesn't change during program execution
** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...
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 ''L''
2''(G/K)'' of ''G'' is multiplicity free i.e. it is a
direct integral In mathematics and functional analysis a direct integral is a generalization of the concept of direct sum. The theory is most developed for direct integrals of Hilbert spaces and direct integrals of von Neumann algebras. The concept was introduced i ...
of distinct
unitary
Unitary may refer to:
Mathematics
* Unitary divisor
* Unitary element
* Unitary group
* Unitary matrix
* Unitary morphism
* Unitary operator
* Unitary transformation
* Unitary representation
* Unitarity (physics)
* ''E''-unitary inverse semigroup ...
irreducible representations.
For a classification of such Gelfand pairs see.
[O. Yakimova]
Gelfand pairs
PhD thesis submitted to Bonn University.
Classical examples of such Gelfand pairs are ''(G,K)'', where ''G'' is a
reductive Lie group
In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direct ...
and ''K'' is a
maximal compact subgroup In mathematics, a maximal compact subgroup ''K'' of a topological group ''G'' is a subgroup ''K'' that is a compact space, in the subspace topology, and maximal amongst such subgroups.
Maximal compact subgroups play an important role in the classi ...
.
Locally compact topological group with compact subgroup
When ''G'' is a
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
topological group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two str ...
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
In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest ...
continuous
measures
Measure may refer to:
* Measurement, the assignment of a number to a characteristic of an object or event
Law
* Ballot measure, proposed legislation in the United States
* Church of England Measure, legislation of the Church of England
* Meas ...
on ''G'' with multiplication defined by convolution is commutative.
* For any
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
locally convex
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ve ...
irreducible representation
In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W,W ...
''π'' of ''G'', the space ''π''
''K'' of ''K''-
invariant
Invariant and invariance may refer to:
Computer science
* Invariant (computer science), an expression whose value doesn't change during program execution
** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...
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 ''L''
2''(G/K)'' of ''G'' is multiplicity free i.e. it is a
direct integral In mathematics and functional analysis a direct integral is a generalization of the concept of direct sum. The theory is most developed for direct integrals of Hilbert spaces and direct integrals of von Neumann algebras. The concept was introduced i ...
of distinct
unitary
Unitary may refer to:
Mathematics
* Unitary divisor
* Unitary element
* Unitary group
* Unitary matrix
* Unitary morphism
* Unitary operator
* Unitary transformation
* Unitary representation
* Unitarity (physics)
* ''E''-unitary inverse semigroup ...
irreducible representations.
In that setting, ''G'' has an
Iwasawa-
Monod Monod is a surname, and may refer to:
* Adolphe Monod (1802–1856), French Protestant churchman; brother of Frédéric Monod.
* Frédéric Monod (1794–1863), French Protestant pastor.
* Gabriel Monod, French historian
* Jacques Monod (1910–19 ...
decomposition, namely ''G = K P'' for some
amenable subgroup ''P'' of ''G''.
Nicolas Monod
Nicolas Monod is a professor at École Polytechnique Fédérale de Lausanne (EPFL) and known for work on bounded cohomology, ergodic theory, geometry ( CAT(0) spaces), locally compact groups and amenability.
He was born in Montreux, Switzerla ...
, "Gelfand pairs admit an Iwasawa decomposition". This is the abstract analogue of the
Iwasawa decomposition In mathematics, the Iwasawa decomposition (aka KAN from its expression) of a semisimple Lie group generalises the way a square real matrix can be written as a product of an orthogonal matrix and an upper triangular matrix (QR decomposition, a conseq ...
of
semisimple Lie group
In mathematics, a Lie algebra is semisimple if it is a direct sum of modules, direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper Lie algebra#Subalgebras.2C ideals and homomorphisms, i ...
s.
Lie group with closed subgroup
When ''G'' is a
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
and ''K'' is a
closed subgroup
In mathematics, topological groups are logically the combination of Group (mathematics), groups and Topological space, topological spaces, i.e. they are groups and topological spaces at the same time, such that the Continuous function, continui ...
, the pair ''(G,K)'' is called a generalized Gelfand pair if for any irreducible
unitary representation In mathematics, a unitary representation of a group ''G'' is a linear representation π of ''G'' on a complex Hilbert space ''V'' such that π(''g'') is a unitary operator for every ''g'' ∈ ''G''. The general theory is well-developed in case ''G'' ...
''π'' of '' G'' on a
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
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
In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direct ...
over a
local field
In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact t ...
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
, where
denotes the
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebraic ...
dual
Dual or Duals may refer to:
Paired/two things
* Dual (mathematics), a notion of paired concepts that mirror one another
** Dual (category theory), a formalization of mathematical duality
*** see more cases in :Duality theories
* Dual (grammatical ...
.
GP3) For any irreducible
unitary representation In mathematics, a unitary representation of a group ''G'' is a linear representation π of ''G'' on a complex Hilbert space ''V'' such that π(''g'') is a unitary operator for every ''g'' ∈ ''G''. The general theory is well-developed in case ''G'' ...
''π'' of ''G'' on a
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
the dimension of Hom
''K''(''π'', C) is less than or equal to 1.
Here, ''admissible representation'' is the usual notion of
admissible representation
In mathematics, admissible representations are a well-behaved class of representations used in the representation theory of reductive Lie groups and locally compact totally disconnected groups. They were introduced by Harish-Chandra.
Real or com ...
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.
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: . Sometimes, this property is also called the multiplicity one property.
In 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
In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W,W ...
''π'' of ''G'' and ''τ'' of ''K'', the space Hom
''K''(''τ'',''π'') is no-more-than-1-dimensional.
* For any irreducible representation ''π'' of ''G'' and ''τ'' of ''K'', the space Hom
''K''(''π'',''τ'') is no-more-than-1-dimensional.
Criteria for Gelfand property
Locally compact topological group with compact subgroup
In this case there is a classical criterion due to
Gelfand ''Gelfand'' is a surname meaning "elephant" in the Yiddish language and may refer to:
* People:
** Alan Gelfand, the inventor of the ollie, a skateboarding move
** Alan E. Gelfand, a statistician
** Boris Gelfand, a chess grandmaster
** Israel Gel ...
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 ''Gelfand'' is a surname meaning "elephant" in the Yiddish language and may refer to:
* People:
** Alan Gelfand, the inventor of the ollie, a skateboarding move
** Alan E. Gelfand, a statistician
** Boris Gelfand, a chess grandmaster
** Israel Gel ...
and
Kazhdan for the pair ''(G,K)'' to satisfy GP2. Suppose that there exists an
involutive anti
Anti may refer to:
*Anti-, a prefix meaning "against"
*Änti, or Antaeus, a half-giant in Greek and Berber mythology
*A false reading of ''Nemty'', the name of the ferryman who carried Isis to Set's island in Egyptian mythology
* Áńt’į, or ...
-
automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
''σ'' of ''G'' such that any ''(K,K)''-double invariant
distribution Distribution may refer to:
Mathematics
*Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations
* Probability distribution, the probability of a particular value or value range of a vari ...
on ''G'' is ''σ''-invariant. Then the pair ''(G,K)'' satisfies GP2. See.
Israel Gelfand
Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gel'fand, or Izrail M. Gelfand ( yi, ישראל געלפֿאַנד, russian: Изра́иль Моисе́евич Гельфа́нд, uk, Ізраїль Мойсейович Гел ...
, David Kazhdan
David Kazhdan ( he, דוד קשדן), born Dmitry Aleksandrovich Kazhdan (russian: Дми́трий Александро́вич Кажда́н), is a Soviet and Israeli mathematician known for work in representation theory. Kazhdan is a 1990 Ma ...
, 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_(F),GL_n(F)) is a Gelfand pair for any local field F. ]
If the above statement holds only for
positive definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular:
* Positive-definite bilinear form
* Positive-definite f ...
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
Anti may refer to:
*Anti-, a prefix meaning "against"
*Änti, or Antaeus, a half-giant in Greek and Berber mythology
*A false reading of ''Nemty'', the name of the ferryman who carried Isis to Set's island in Egyptian mythology
* Áńt’į, or ...
-
automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
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
Anti may refer to:
*Anti-, a prefix meaning "against"
*Änti, or Antaeus, a half-giant in Greek and Berber mythology
*A false reading of ''Nemty'', the name of the ferryman who carried Isis to Set's island in Egyptian mythology
* Áńt’į, or ...
-
automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
''σ'' of '' G'' s.t. any ''K'' × ''K'' invariant positive definite
distribution Distribution may refer to:
Mathematics
*Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations
* Probability distribution, the probability of a particular value or value range of a vari ...
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 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 case when ''K'' is compact it meanes that the dimension of Hom
K(π, χ)) is less than or equal to 1. One can adapt the criterion for Gelfand pairs 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
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
''θ'' of '' G'' such that ''K'' is a union of connected components of the group of ''θ''-invariant elements: ''G''
''θ''.
If ''G'' is a
connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
reductive group
In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direct ...
over R and ''K=G''
''θ'' is a compact
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a 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
In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact t ...
has the Gelfand property. For symmetric pairs of rank one this question was investigated in
[G. van Dijk. On a class of generalized Gelfand pairs, Math. Z. 193, 581-593 (1986).] and
An example of high rank Gelfand symmetric pair is (GL(''n+k''), GL(''n'') × GL(''k'')). This was proven in
Hervé Jacquet
Hervé Jacquet is a French American mathematician, working in automorphic forms. He is considered one of the founders of the theory of automorphic representations and their associated L-functions, and his results play a central role in modern num ...
, Stephen Rallis
Uniqueness of linear periods.
Compositio Mathematica , tome 102, n.o. 1 , p. 65-123 (1996). over non-archimedean local fields and later in
[A. Aizenbud, D. Gourevitch, An archimedean analog of Jacquet - Rallis theorem. ] for all local fields of
characteristic zero.
For more details on this question for high rank symmetric pairs see.
[A. Aizenbud, D.Gourevitch, Generalized Harish-Chandra descent and applications to Gelfand pairs. ]
Spherical pairs
In the context of algebraic groups the analogs of Gelfand pairs are called
spherical pair. 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
In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' invertible matrices), the subgroup ...
of ''G''.
* There is a finite number of ''(B,K)''-double coset in ''G''
* For any algebraic representation ''π'' of ''G'', we have dim
.
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 ''π'' of ''G'', the space Hom
''K''(''π'',C) is finite-dimensional. Moreover, the bound for this dimension does not depend on ''π''. This conjecture is proven for a large class of spherical pairs including all the
symmetric pairs.
[Yiannis Sakellaridis and ]Akshay Venkatesh
Akshay Venkatesh (born 21 November 1981) is an Australian mathematician and a professor (since 15 August 2018) at the School of Mathematics at the Institute for Advanced Study. His research interests are in the fields of counting, equidistribu ...
, "Periods and harmonic analysis on spherical varieties".
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 called ''K''-distinguished if Hom
''K''(''π'',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
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ce ...
, 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
In mathematics, specifically in the representation theory of Lie groups, a Harish-Chandra module, named after the Indian mathematician and physicist Harish-Chandra, is a representation of a real Lie group, associated to a general representation, w ...
.
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,
Joseph Shalika
Joseph Andrew Shalika (June 25, 1941 – September 18, 2010) was a mathematician working on automorphic forms and representation theory, who introduced the multiplicity-one theorem. He was a member of the Institute for Advanced Study
The ...
, The multiplicity one theorem for GL''n'', Ann. of Math. 100(1974) 171–193. ). In this case ''K''-distinguished representations are called generic
Generic or generics may refer to:
In business
* Generic term, a common name used for a range or class of similar things not protected by trademark
* Generic brand, a brand for a product that does not have an associated brand or trademark, other ...
(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 representation theory, a branch of mathematics, the Whittaker model is a realization of a representation of a reductive algebraic group such as ''GL''2 over a finite or local or global field on a space of functions on the group. It is named aft ...
.
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,[Omer Offen, Eitan Sayag, Global Mixed Periods and local Klyachko models for the general linear group, ][Omer Offen, Eitan Sayag, UNIQUENESS AND DISJOINTNESS OF KLYACHKO MODELS , ])
Gelfand–Zeitlin construction
One can also use Gelfand pairs for constructing bases for irreducible representations: suppose we have a sequence ⊂ ''G1'' ⊂ ... ⊂ ''Gn'' s.t. ''(Gi,Gi-1)'' is a strong Gelfand pair. For simplicity let's assume that ''Gn'' is compact. Then this gives a canonical decomposition of any irreducible representation of ''Gn'' to one-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 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 In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields:
* Algebraic number field: A finite extension of \mathbb
*Global function fi ...
''F'' and let ''K'' be an algebraic subgroup of ''G''. Suppose that for any place
Place may refer to:
Geography
* Place (United States Census Bureau), defined as any concentration of population
** Census-designated place, a populated area lacking its own Municipality, municipal government
* "Place", a type of street or road ...
of ''F'' the pair (''G'', ''K'') is a Gelfand pair over the completion . Let ''m'' be an automorphic form
In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of ...
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 ).
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
In mathematics, an ''L''-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. An ''L''-series is a Dirichlet series, usually convergent on a half-plane, that may give ris ...
and this gives an analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new ...
and functional equation
In mathematics, a functional equation
is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...
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
Harmonic analysis is a branch of mathematics concerned with the representation of Function (mathematics), functions or signals as the Superposition principle, superposition of basic waves, and the study of and generalization of the notions of Fo ...
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
In mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if ''M'' and ''N'' are two finite-dimensional irreducible representations
of a group ' ...
.
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
Relative may refer to:
General use
*Kinship and family, the principle binding the most basic social units society. If two people are connected by circumstances of birth, they are said to be ''relatives''
Philosophy
*Relativism, the concept that ...
is obtained from the trace formula by this procedure.
Examples
Finite groups
A few common examples of Gelfand pairs are:
* (Sym(''n''+1), Sym(''n'')), the symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \m ...
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
In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
.
If ''(G,K)'' is a Gelfand pair, then (''G''/''N'',''K''/''N'') is a Gelfand pair for every ''G''-normal subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G i ...
''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 for every ''χ'' in Irr(''G''). Since