Gelfand pair

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In mathematics, a pair (G,K) consisting of a locally compact group G which is unimodular, and a subgroup K which is a compact group, is a Gelfand pair when, roughly speaking, the (K,K)-double cosets in G commute. More precisely, the Hecke ring, the algebra of compactly supported functions on G that are invariant under translation on either side by K, should be commutative for the convolution on G (for Haar measure). The abstract theory of Gelfand pairs (of Israel Gelfand) is closely related to classical topics on spherical functions and Riemannian symmetric spaces.

Then the algebra L¹(K\G/K) is a commutative Banach algebra. Its characters give rise to the zonal spherical functions on G. These include, in special cases, Legendre functions and Bessel functions.

[edit] Construction

Many Gelfand pairs arise via the following general construction. G is a locally compact group with an involution

$\theta:G\to G$

and K is the subgroup of its fixed points. Then by some general reasoning, $θ$ induces a map of the Hecke ring into itself which is both an involution and anti-involution. Therefore, the Hecke ring is commutative.

[edit] Examples

1. If G is a semisimple Lie group and K is its maximal compact subgroup, then Elie Cartan's theory of Riemannian symmetric spaces implies the existense of the involution $θ$ with the requisite properties, the Cartan involution. Therefore, (G,K) is a Gelfand pair. More specifically, let G=SL_n(R) be the real special linear group and $θ(g) = (g *) - 1,$ where the star denotes the conjugate transpose of a matrix. Then $θ$ is an involution whose fixed point set is the special unitary group SU_n(R), which is a maximal compact subgroup of G.

2. Any Riemannian symmetric space leads to a Gelfand pair. For example, let G=SU_n(R) and $θ(g) = (g t) - 1$ be the inverse transpose of a matrix. Then K=SO_n(R) is the special orthogonal subgroup, G/K is a symmetric space of compact type, and (SU_n(R), SO_n(R)) is a Gelfand pair.

3. Let G=SL_n(C) and K=SL_n-1(C) be the subgroup stabilizing the first n-1 basis vectors. Then these groups form a Gelfand pair, with $θ$ given by conjugation by the diagonal matrix with 1 in first n-1 places and -1 in the last place.

[edit] Significance in representation theory

See also: Zonal spherical function

Very general considerations imply that if (G,K) is a Gelfand pair then the subspace of K-invariants in any irreducible representation of G is at most one-dimensional. This directly leads to the theory of spherical functions.

If G is a complex linear algebraic group and $θ$ is a complex linear map obtained by complexification of a Cartan involution of a real form of G, as in Example 3 above, then the Gelfand pair arising by the construction of the previous section has a stronger property that K is a spherical subgroup of G. This implies that any irreducible reperesentation of G decomposes simply (i.e. without multiplicities) when restricted to K. This is a very rare and distinguished situation in representation theory which greatly simplifies analysis of irreducible representations.