Spherical function on a Gelfand pair - Imported Knowlpedia Content

Definition

Let $G$ be a locally compact group and $K\leq G$ a compact subgroup such that the algebra of compactly supported continuous $K$ -bi-invariant functions is commutative under convolution; then $(G,K)$ is a Gelfand pair. A spherical function is a nonzero continuous $K$ -bi-invariant function $\varphi:G\to\mathbb C$ , normalized by $\varphi(e)=1$ , that satisfies

\int_K\varphi(xky)\,dk=\varphi(x)\varphi(y) \qquad(x,y\in G),

where $dk$ is normalized Haar measure on $K$ . Equivalently, integration against $\varphi$ defines a nonzero multiplicative functional on this test-function convolution algebra. The normalization excludes nontrivial scalar rescalings and makes the product formula canonical.

Representation-theoretic realization

If $\pi$ is an irreducible spherical representation and $v$ is a unit $K$ -fixed vector, then

\varphi_\pi(g)=\langle\pi(g)v,v\rangle

is a positive-definite spherical function. Conversely, every positive-definite spherical function arises this way, up to unitary equivalence. This identifies the positive-definite spherical spectrum with the $K$ -spherical part of the unitary dual Helgason, Chapter IV, §§2–3.

Convolution eigenfunctions

For every integrable $K$ -bi-invariant $f$ , convolution by $f$ acts on a spherical function by a scalar determined by the spherical transform. This simultaneous diagonalization is the commutative harmonic analysis of the homogeneous space $G/K$ ; the product formula in the core is precisely what makes all invariant convolution operators share these eigenfunctions.

Conventions and scope

Some sources build positive definiteness into “zonal spherical function,” while others call every normalized convolution-algebra character “elementary spherical.” The core adopts the broader algebraic convention, so a spherical function need not be unitary or positive-definite. The compact subgroup $K$ and its normalized Haar measure are part of the setting.

References

Sigurdur Helgason, Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions, American Mathematical Society, 2000. AMS record. Relevant: Chapter IV, §§2–3 on spherical functions and class-one representations.
Roger Godement, “A Theory of Spherical Functions. I,” Transactions of the American Mathematical Society 73 (1952), 496–556. DOI record. Relevant: the abstract theory of spherical functions and Gelfand pairs.