Abstract harmonic analysis Homogeneous spaces, Gelfand pairs and spherical functions Theorem

Paley–Wiener theorem for symmetric spaces

Builds on Semisimple Lie group Maximal compact subgroup Cartan decomposition of a Lie group Smooth function Entire function Spherical transform Harish-Chandra's formula for spherical functions

Statement

Let $G$ be a connected noncompact semisimple Lie group with finite centre and $K$ a maximal compact subgroup. Let $\mathfrak g = \mathfrak k \oplus \mathfrak p$ be the Cartan decomposition, $\mathfrak a \subseteq \mathfrak p$ a maximal abelian subspace, and $W$ the Weyl group of the restricted roots. Give $\mathfrak p$ and $\mathfrak a^*$ the norms $\abs{\cdot}$ induced by the Killing form, and for $\lambda \in \mathfrak a^*_\C$ let $\abs{\operatorname{Im} \lambda}$ be the norm of its imaginary part. By the Cartan decomposition each $g \in G$ is $g = k \exp X$ with $k \in K$ and $X \in \mathfrak p$ unique; put $\sigma(g) = \abs{X}$ . Let $\hat f(\lambda) = \int_G f(g) \varphi_{-\lambda}(g) \, dg$ be the spherical transform, with $\varphi_\lambda$ given by Harish-Chandra's formula. Fix $R > 0$ . Let $\mathcal{D}_R$ be the space of smooth bi- $K$ -invariant functions $f$ on $G$ with support in $\set{g : \sigma(g) \le R}$ . Let $\mathcal{E}_R$ be the space of $W$ -invariant entire functions $F$ on $\mathfrak a^*_\C$ such that, for every integer $N \ge 0$ , $\sup_{\lambda \in \mathfrak a^*_\C} \, (1 + \abs{\lambda})^N \, e^{-R \abs{\operatorname{Im} \lambda}} \, \abs{F(\lambda)} < \infty.$ Then $f \mapsto \hat f$ is a bijection from $\mathcal{D}_R$ onto $\mathcal{E}_R$ .

Remarks

The function $\sigma(g)$ is the Riemannian distance from the origin $K$ to $g K$ in $X = G / K$ , for the metric given by the Killing form. So the support condition says that $f$ lives on the closed ball of radius $R$ . That $\hat f$ has this growth follows from Harish-Chandra's formula and integration by parts with the Laplacian. The converse uses the inversion formula of the Plancherel theorem for spherical functions and a contour shift, where the explicit form of the c-function controls the poles of $\abs{c(\lambda)}^{-2} = (c(\lambda) c(-\lambda))^{-1}$ . The theorem is due to Helgason and Gangolli; Helgason later proved a version for the Helgason Fourier transform of functions that are not radial.

Example: for $G = \R^n \rtimes \mathrm{SO}(n)$ and $K = \mathrm{SO}(n)$ , which is not semisimple but has the same structure, the analogous statement is the classical Paley–Wiener theorem for radial functions on $\R^n$ .

Going deeper

For a student

The theorem identifies the image of $C_c^\infty(K \backslash G / K)$ under the spherical transform, and with it the convolution algebra of compactly supported radial functions as an algebra of entire functions. Helgason used it to prove that invariant differential operators on $X$ are surjective on smooth functions. In rank one the Weyl group is $\set{\pm 1}$ , and $W$ -invariance is the evenness familiar from the radial Fourier transform on $\R$ .

Tag	`QA8BFW2QS4`
URI	`m2://QA8BFW2QS4`
Kind	statement, theorem
Writing	W0 stub
MSC	43A85, 43A90, 22E30
Status	This copy of the site has no build data. Run `m2 status QA8BFW2QS4`.
About	Spherical transform
Uses	Semisimple Lie group, Maximal compact subgroup, Cartan decomposition of a Lie group, Smooth function, Entire function, Spherical transform, Harish-Chandra's formula for spherical functions
Changed	3 November 2026

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