Talk:Gelfand pair

Importance?
Why are Gelfand pairs important? Commutative Schur rings are discussed by Wielandt in his Finite Permutation Groups text, but he doesn't directly mention their importance. They seem to be often studied, but why? JackSchmidt (talk) 15:39, 27 August 2008 (UTC)

Hi, hier are sum aplications of Gelfand pairs if u can cline them and put them in the artical I'll b gretfull

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 $$ \dim( \operatorname{Hom}_K(\pi, \mathbb{C}) ) = 1 $$. The representation $$ Ind_G^K(\Complex) $$ 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. Examples:

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 Harish-Chandra module.

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 $$ \psi$$ (see, ). In this case K-distinguished representations are called generic and they are easy to classify. Almost any irreducible representation is generic. The unique (up to scalar) imbedding of a generic representation to $$Ind_G^K(\psi)$$ is called whitaker model.

In case G=GL(n) there is a finer version of the result above, namely there exist a finite sequence of subgroups K_i and characters $$ \psi_i$$ s.t. (G,K_i) is twisted Gelfand pair w.r.t. $$\psi_i$$ and any irreducible unitary representation is  K_i distinguished for exactly one i (see, )

Gelfand - Zeitlin construction
Another use of Gelfand pairs is for construction of bases of irreducible representations. Suppose that we have a sequence $$ {1} \subset G_1 \subset...\subset G_n$$ 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 one dimensional subrepresentations. For the case $$ G_n=U_n $$ (the unitary group) this construction is called Gelfand Zeitlin basis. Note that 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 $$GL_n$$.

Remark: this basis isn't canonical as it depends of the choice of the embeddings $$ U_i \subset U_{i+1}$$

Splitting of periods of automorphic forms
more modern 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 its algebraic subgroup. Suppose that for any place $$ \nu$$  of F the pair (G,K) is a Gelfand pair over the completion $$F_\nu$$. Let m be an automorphic form over G, then its H-period splits to a produced of local factors (i.e. factors that depends only on the behavior of m at each place$$ \nu$$).

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 $$G \times G$$. Indeed, to know all the irreducible representations of G is equivalent to know the decomposition of the space of functions on G as a $$G \times G$$ representation. In this approach representation theory can be generalized by replacing the pair $$(G \times G, G) $$ by any spherical pair (G,K). Then we will be lead 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.

Extremal pairs for finite groups
So the obvious Gelfand pair is (G, G), but this is pretty silly. However, the larger the subgroup, the easier it is for it to be a Gelfand pair. In particular, one expects maximal subgroups to have a good chance of forming Gelfand pairs. For groups of order at most 659, all maximal subgroups form Gelfand pairs, and for primitive groups of degree at most 54, the point stabilizer forms a Gelfand pair. However, PSL(2,11) acting on its subgroup D12 is primitive of degree 55 and of order 660, but is not a Gelfand pair.

At least when K is G-corefree, if K ≤ U ≤ G and (G, K) is a Gelfand pair, then so is (G, U) and (U, K). In particular then, maximal subgroups that *are* Gelfand pairs are fairly dull. What we really want are maximal subgroups that are not Gelfand, and then a list of subgroups minimal with respect to being Gelfand. Anyone happen to know of a reference for these? JackSchmidt (talk) 17:26, 27 August 2008 (UTC)
 * The p for which every maximal subgroup M defines a Gelfand pair (PSL(2,p),M) are known, . There are some other interesting articles on Gelfand pairs of classical groups as well, but the results are less easily summarized.  Focus is given to interesting pairs, and then deciding how close to Gelfand they are, rather than finding pairs based on their Gelfandian qualities.  Sp(4,2^(2n+1)), Sz(2^(2n+1)) is a nice Gelfand pair, but I haven't decided if it is part of a larger pattern yet,.
 * JackSchmidt (talk) 21:56, 28 August 2008 (UTC)
 * JackSchmidt (talk) 21:56, 28 August 2008 (UTC)
 * JackSchmidt (talk) 21:56, 28 August 2008 (UTC)


 * Also looking for a reference for the fact that if G is finite solvable and M is maximal, then (G,M) is a Gelfand pair. The proof is not too hard, but combines material from a few sources nontrivially. JackSchmidt (talk) 19:24, 29 August 2008 (UTC)

Some proposels
1) I propose to add a section Operetions with Gelfand pairs (may b under a diferent name e.g. General remarks). And then there will b no need in the last item in the definition of gelfand pair of finet groups, wich seems not to fit under the title definition.

2) It seems that there is quete big amaunt of matirial on gelfand pairs of finet groups, some of it is wreten in the artical in the section "exampals" and some in the discution page. It dosen't seems to fit the title "exampals" as it more thet jast exampals. I think that it deserves its own page or subpage, and a link to it in the sub section on the definition of gelfand pair of finet groups. If you dis-egry then what abut a seperate section insted of subsection in the section "exampals"?

what do u think Aizenr (talk) 14:07, 6 November 2008 (UTC)