Gelfand pair - Imported Knowlpedia Content

Definition

Let $G$ be a locally compact group and $K\leq G$ a compact subgroup. The pair $(G,K)$ is a Gelfand pair if the convolution algebra

C_c(K\backslash G/K) =\{f\in C_c(G):f(k_1gk_2)=f(g)\}

is commutative, where convolution is formed using a left Haar measure on $G$ . Equivalently, the $K$ -bi-invariant part of $L^1(G)$ is commutative. The definition depends on the pair, not on $G$ alone, and equips the homogeneous space $G/K$ with a commutative spherical harmonic analysis.

Representation-theoretic characterization

When $G$ is second countable and type I, $(G,K)$ is a Gelfand pair exactly when the quasi-regular representation of $G$ on $L^2(G/K)$ is multiplicity-free. Equivalently, every irreducible unitary representation of $G$ has at most one $K$ -fixed vector. These equivalences connect commutativity of convolution with the one-dimensional spherical eigenspaces that underlie spherical functions Folland, §9.5.

Examples and non-examples

For $n\geq 2$ , the Euclidean motion pair $(\mathbb R^n\rtimes \mathrm{SO}(n),\mathrm{SO}(n))$ is a Gelfand pair; its spherical analysis reduces to radial Fourier analysis. Compact symmetric pairs, such as $(\mathrm{SO}(n+1),\mathrm{SO}(n))$ , give another basic family.

If $G$ is a nonabelian discrete group and $K=\{e\}$ , then $C_c(K\backslash G/K)=C_c(G)$ is the noncommutative group convolution algebra. Thus $(G,\{e\})$ is not a Gelfand pair, although $G/\{e\}$ is still a homogeneous space.

Conventions and scope

Some authors use a dense test-function algebra, the $K$ -bi-invariant part of $L^1(G)$ , or the convolution algebra of compactly supported $K$ -bi-invariant measures. For compact $K$ these standard formulations agree. “Commutative homogeneous space” is shorthand for the pair $(G,K)$ ; it does not assert that the space $G/K$ itself carries a group law.

References

Gerald B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. DOI record. Relevant: §9.5 on Gelfand pairs, spherical functions, and multiplicity.
Jacques Faraut, Analysis on Lie Groups: An Introduction, Cambridge University Press, 2008. Publisher record. Relevant: Chapter 9 on spherical analysis of the sphere and Euclidean space.