Spherical Plancherel theorem - Imported Knowlpedia Content

Statement

Let $G$ be a connected noncompact semisimple Lie group with finite center, $K$ a maximal compact subgroup, and $X=G/K$ . For a $K$ -invariant $f\in L^1(X)\cap L^2(X)$ , define

\widehat f(\lambda)=\int_X f(x)\varphi_{-\lambda}(x)\,dx ,

where $\varphi_\lambda$ is the normalized spherical function. Choose compatible Haar and Lebesgue measures, and let $d\bar\lambda$ denote the pushforward to $\mathfrak a^*/W$ of $|W|^{-1}d\lambda$ , for the chosen Lebesgue measure on $\mathfrak a^*$ . The spherical Plancherel theorem says that this transform extends to a unitary map

L^2(X)^K\longrightarrow L^2\!\left(\mathfrak a^*/W,\, |c(\lambda)|^{-2}d\bar\lambda\right),

where $c$ is the Harish–Chandra c-function. Equivalently, the transform preserves inner products and admits spherical Fourier inversion in the $L^2$ -sense.

Inversion formula

For sufficiently regular rapidly decreasing $K$ -invariant $f$ , Fourier inversion takes the pointwise form

f(x)=C\int_{\mathfrak a^*} \widehat f(\lambda)\varphi_\lambda(x) |c(\lambda)|^{-2}\,d\lambda ,

where $C$ is determined by the chosen normalizations and includes the Weyl group convention. Density then gives the unitary extension to all of $L^2(X)^K$ . The theorem is the commutative, multiplicity-one part of the nonabelian Plancherel decomposition Helgason, Chapter IV, §7.

Interpretation

The commuting algebra of $G$ -invariant differential operators on $X$ is simultaneously diagonalized by the functions $\varphi_\lambda$ . The factor $|c(\lambda)|^{-2}$ is therefore not an arbitrary weight: it is the spectral density forced by the asymptotics of those joint eigenfunctions. Passing to $\mathfrak a^*/W$ removes the redundancy $\varphi_{w\lambda}=\varphi_\lambda$ .

Examples and scope

For Euclidean space, viewed through its motion-group Gelfand pair, the spherical transform of radial functions becomes the Fourier–Bessel transform. On a real-rank-one noncompact symmetric space, the spectral parameter is one-dimensional and the density is an explicit quotient of gamma factors.

This statement concerns the $K$ -invariant subspace of $L^2(G/K)$ , not the full Plancherel decomposition of $L^2(G)$ . General $K$ -types require matrix-valued transforms and generalized $c$ -functions.

References

Sigurdur Helgason, Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions, American Mathematical Society, 2000. AMS record. Relevant: Chapter IV, §7 on spherical Fourier inversion and the Plancherel formula.
Ramesh Gangolli and V. S. Varadarajan, Harmonic Analysis of Spherical Functions on Real Reductive Groups, Springer, 1988. Publisher record. Relevant: the spherical transform, wave packets, and Plancherel theory on real reductive groups.