Gelfand pair

In mathematics, the expression Gelfand pair refers to a pair ("G", "K") consisting of a group "G" and a subgroup "K" that satisfies a certain property on restricted representations.

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 "H" of any irreducible representation of "G" contains the trivial representation of H with multiplicity no more than 1. In each case one should specify the class of considered representations and the meaning of contains.

The theory of Gelfand pairs is closely related to the topics spherical functions and Riemannian symmetric spaces.

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

In 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", we have $dim(\; operatorname\{Hom\}\_K(pi,\; mathbb\{C\})\; )\; leq\; 1$, where $mathbb\{C\}$ 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".

Lie group with compact subgroup

In case when "G" is a Lie group and "K" is 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" we have $Hom\_K(pi,\; mathbb\{C\})\; leq\; 1$.

For a classification of such Gelfand pairs see O. Yakimova. [http://hss.ulb.uni-bonn.de/diss_online/math_nat_fak/2004/yakimova_oksana/yakimova-abstract-engl.htm Gelfand pairs] , PhD thesis submitted to Bonn university. ]

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

In case when "G" is a locally compact topological group and "K" is 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 $pi^K$ of "K"-invariant vectors in "π" is no more than 1 dimensional.
* For any continuous, locally convex, irreducible representation "π" of "G", we have $dim(\; Hom\_K(pi,\; mathbb\{C\})\; )\; leq\; 1$.

Lie group with closed subgroup

In case when "G" is a Lie group and "K" is 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 we have $Hom\_K(pi^\{infty\},\; mathbb\{C\})\; leq\; 1$, where $pi^\{infty\}$ denotes the subrepresentation of smooth vectors.

Reductive group over a local field with closed subgroup

In case when "G" is a reductive group over a local field and "K" is 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 admissible representation "π" of "G" we have $dim\; Hom\_K(pi,\; mathbb\{C\})\; leq\; 1$

GP2) For any admissible representation "π" of "G" we have $dim\; Hom\_K(pi,\; mathbb\{C\})\; cdot\; dim\; Hom\_K(\; ilde\{pi\},\; mathbb\{C\})\; leq\; 1$, where $ilde\{pi\}$ denotes the smooth dual.

GP3) For any irreducible unitary representation "π" of "G" on a Hilbert space we have $dim\; Hom\_K(pi^\{infty\},\; mathbb\{C\})\; leq\; 1$

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.

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, Δ"K" = { ( "k", "k" ) in "G" × "K" : "k" in "K" }. 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 "π" of "G" and "τ" of "K", the space $Hom\_K(\; au,\; pi)$ is no more than 1 dimensional.
* For any irreducible representation "π" of "G" and "τ" of "K", the space $Hom\_K\; (pi,\; au)$ 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 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 I.M. Gelfand, D. Kazhdan, Representations of the group GL(n,K) where K is a local field, Lie groups and their representations (Proc. Summer School, Bolyai Janos Math. Soc., Budapest, 1971), pp. 95--118. Halsted, New York (1975). ] and A. Aizenbud, D. Gourevitch, E. Sayag : (GL_{n+1}(F),GL_n(F)) is a Gelfand pair for any local field F. [http://arxiv.org/abs/0709.1273 http://arxiv.org/abs/0709.1273] ]

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 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 $chi$ of a the grope K, if the Gelfand property holds true when the trivial representation is replaced with the character $chi$. For example in case when K is compact it meanes that $dim(\; operatorname\{Hom\}\_K(pi,\; chi)\; )\; leq\; 1$ . 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 "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 reductive group over $mathbb\{R\}$ and $K=\; G^\; heta$ 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 G. van Dijk. On a class of generalized Gelfand pairs, Math. Z. 193, 581-593 (1986). ] and E. P. H. Bosman and G. Van Dijk, A New Class of Gelfand Pairs, Geometriae Dedicata 50, 261-282, 261 Kluwer Academic Publishers. Printed in the Netherlands (1994).]

An example of high rank Gelfand symmetric pair is $(GL\_\{n+k\},\; GL\_n\; imes\; GL\_k)$. This was proven in H. Jacquet, S. Rallis, [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 , p. 65-123 (1996). ] over non-archimedean local fields and later in A. Aizenbud, D. Gourevitch, An archimedean analog of Jacquet - Rallis theorem. [http://arxiv.org/abs/0803.3397 http://arxiv.org/abs/0709.1273] ] 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. [http://arxiv.org/abs/0803.3395 http://arxiv.org/abs/0803.3395] ] .

Spherical pairs

If "G" is a reductive group over a local field there is another property that is weaker than Gelfand property, but is easier to verify. Namely, the pair ("G","K") is called a spherical pair if one the following equivalent conditions holds.

* For any parabolic subgroup "P" of "G" there exists an open ("P","K")-double coset in "G".
* For any parabolic subgroup "P" of " G" there is a finite number ("P","K")-double cosets in "G".
* For any admissible representation "π" of "G" $Hom\_K(pi,\; mathbb\{C\})$ is finite dimensional.

Examples

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 $[1\_K,chidownarrow^G\_K]\; leq\; 1$ for every "χ" in Irr("G"). Since $[1\_K,chidownarrow^G\_K]\; =\; [1uparrow\_K^G,chi]$ by Frobenius reciprocity and $1uparrow\_K^G$ 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 harv|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, harv|Isaacs|1994|p=69.

Indeed, if "G" is a transitive permutation group whose point stabilizer "K" has at most 4 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 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"

* $(GL\_n(mathbb\{R\})\; ,\; O\_n(mathbb\{R\}))$
* $(GL\_n(mathbb\{C\})\; ,\; U\_n)$
* $(O\_\{n+k\}(mathbb\{R\})\; ,\; O\_n(mathbb\{R\})\; imes\; O\_k(mathbb\{R\}))$
* $(U\_\{n+k\}\; ,\; U\_n\; imes\; U\_k)$
* ("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 $n\; geq\; 4$
* $(Sp\_\{2n+2\}(F),Sp\_\{2n\}(F)\; imes\; Sp\_\{2\}(F)\; )$ for $n\; geq\; 3$
* $(SO(V),SO(V\; oplus\; F)\; )$ 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:
* $(G\; imes\; G,Delta\; G)$ Follows from Schur's lemma.
* $(GL\_\{n+k\}(F),\; GL\_\{n\}(F)\; imes\; GL\_\{k\}(F))$ See , .
* $(GL\_\{2n\}(F),\; Sp\_\{2n\}(F))$ See Michael J. Heumos and Stephen Rallis. Symplectic-Whittaker models for GLn. Pacific J. Math., 146(2):247–279, 1990. ] and E.Sayag (GL(2n,C),SP(2n,C)) is a Gelfand Pair [http://arxiv.org/abs/0805.2625 http://arxiv.org/abs/0805.2625] ]
* $(O\_\{n+k\}(mathbb\{C\})\; ,\; O\_n(mathbb\{C\})\; imes\; O\_k(mathbb\{C\}))$ See A. Aizenbud, D. Gourevitch. Some regular symmetric pairs. [http://arxiv.org/abs/0805.2504 http://arxiv.org/abs/0805.2504] ] .
* $(GL\_n(mathbb\{C\})\; ,\; O\_n(mathbb\{C\}))$. See
* $(GL\_n(mathbb\{E\})\; ,\; GL\_n(mathbb\{F\}))$, where E is a quadratic extension of F. See Y.Z. Flicker: On distinguished representations, J. Reine Angew. Math. 418 (1991), 139-172. ] , .

Other pairs

The following pairs are strong Gelfand pairs:
* $(S\_\{n+1\}\; ,\; S\_n)$ is a strong Gelfand pair, where $S\_n$ denotes the symmetric group of n elements. This is proven using the involutive anti-automorphism $g\; mapsto\; g^\{-1\}$
* $(GL\_\{n+1\}(F)\; ,\; GL\_n(F))$ where F is a local fields of characteristic zero. See A. Aizenbud, D. Gourevitch, S. Rallis, G. Schiffmann. Multiplicity one Theorems, [http://arxiv.org/abs/0709.4215 http://arxiv.org/abs/0709.4215] ] , A. Aizenbud, D. Gourevitch. Multiplicity one theorem for (GL(n+1,R),GL(n,R)). [http://arxiv.org/abs/0808.2729 http://arxiv.org/abs/0808.2729] ] and B. Sun and C.-B. Zhu [http://www.math.nus.edu.sg/~matzhucb/Multiplicity_One.pdf Multiplicity one theorems: the archimedean case] . ] .
* $(O(V),O(V\; oplus\; F)\; )$ where V is a vector spaceover F with a non- degenerate quadratic form.See and .
* $(U(V),U(V\; oplus\; E)\; )$ where E is a quadratic extension of F and V is a vector space over E with a non- degenerate hermitian form. See and .

Those 4 examples can be rephrased in terms of Gelfand pairs as follows. The pairs

$(S\_\{n+1\}\; imes\; S\_n\; ,\; Delta\; S\_n),\; quad\; (GL\_\{n+1\}(F)\; imes\; GL\_n(F)\; ,\; Delta\; GL\_n(F))\; quad\; (O(V\; oplus\; F)\; imes\; O(V),Delta\; O(V)\; )\; quad\; (U(V\; oplus\; E)\; imes\; U(V),\; Delta\; U(V)\; )$

are Gelfand pairs.

See also

* spherical function
* Riemannian symmetric space
* symmetric pair
* spherical pair

Notes

References

* | year=1996 | journal=Communications in Algebra | issn=0092-7872 | volume=24 | issue=7 | pages=2293–2316
* | year=1994
* | year=1981 | volume=49 | chapter=On multiplicity-free permutation representations | pages=337–353
* | year=1964