Plancherel theorem for spherical functions

In mathematics, the Plancherel theorem for spherical functions is an important result in the representation theory of semisimple Lie groups, due in its final form to Harish-Chandra. It is a natural generalisation in non-commutative harmonic analysis of the Plancherel formula and Fourier inversion formula in the representation theory of the group of real numbers in classical harmonic analysis and has a similarly close interconnection with the theory of differential equations.It is the special case for zonal spherical functions of the general Plancherel theorem for semisimple Lie groups, also proved by Harish-Chandra. The Plancherel theorem gives the eigenfunction expansion of radial functions for the Laplacian operator on the associated symmetric space. In the case of
hyperbolic space, these expansions were known from prior results of Mehler, Weyl and Fock.

History

The first versions of an abstract Plancherel formula for the Fourier transform on a unimodular locally compact group "G" were due to Segal and Mautner. [ harvnb|Helgason|1984|p=492-493, historical notes on the Plancherel theorem for spherical functions] At around the same time, Harish-Chandra [harvnb|Harish-Chandra|1951] [harvnb|Harish-Chandra|1952] and Gelfand & Naimark [harvnb|Gelfand|Naimark|1948] [harvnb|Guillemin|Sternberg|1977] derived an explicit formula for SL(2,R) and complex semisimple Lie groups, so in particular the Lorentz groups. A simpler abstract formula was derived by Mautner for a "topological" symmetric space "G"/"K" corresponding to a maximal compact subgroup "K". Godement gave a more concrete and satisfactory form for positive definite spherical functions, a class of special functions on "G"/"K". Since when "G" is a semisimple Lie group these spherical functions φ_λ were naturally labelled by a parameter λ in the quotient of a Euclidean space by the action of a finite reflection group, it became a central problem to determine explicitly the Plancherel measure in terms of this parametrization. Generalizing the ideas of Hermann Weyl from the spectral theory of ordinary differential equations, Harish-Chandra [ harvnb|Harish-Chandra|1958a] [harvnb|Harish-Chandra|1958b] introduced his celebrated c-function "c"(λ) to describe the asymptotic behaviour of the spherical functions φ_λ and proposed "c"(λ)^–2 "d"λ as the Plancherel measure. He verified this formula for the special cases when "G" is complex or real rank one, thus in particular covering the case when "G"/"K" is a hyperbolic space. The general case was reduced to two conjectures about the properties of the c-function and the so-called spherical Fourier transform. Explicit formulas for the c-function were later obtained for a large class of classical semisimple Lie groups by Bhanu-Murthy. In turn these formulas prompted Gindikin and Karpelevič to derive a product formula [ harvnb|Gindikin|Karpelevič|1962] for the c-function, reducing the computation to Harish-Chandra's formula for the rank 1 case. Their work finally enabled Harish-Chandra to complete his proof of the Plancherel theorem for spherical functions in 1966. [harvnb|Harish-Chandra|1966, section 21]

pherical functions

Let "G" be a semisimple Lie group and "K" a maximal compact subgroup of "G". The Hecke algebra "C"_c("K" "G"/"K"), consisting of compactly supported "K"-biinvariant continuous functions on "G", acts by convolution on the Hilbert space "H"="L"²("G" / "K"). Because "G" / "K" is a symmetric space, this *-algebra is commutative. The closure of its image in the operator norm is a non-unital commutative C* algebra $mathfrak{A}$ , so by the Gelfand isomorphism can be identified with the continuous functions vanishing at infinity on its spectrum "X". [ The spectrum coincides with that of the commutative Banach *-algebra of integrable "K"-biinvariant functions on "G" under convolution, a dense *-subalgebra of $mathfrak{A}$ .] Points in the spectrum are given by continuous *-homomorphisms of $mathfrak{A}$ into C, i.e. characters of $mathfrak{A}$ .

If "S"' denotes the commutant of a set of operators "S" on "H", then $mathfrak{A}^prime$ can be identified with the commutant of the regular representation of "G" on "H". Now $mathfrak{A}$ leaves invariant the subspace "H"₀ of "K"-invariant vectors in "H". Moreover the Abelian von Neumann algebra it generates on "H"₀ is maximal Abelian. By spectral theory, there is an essentially unique [The measure class of μ in the sense of the Radon-Nikodym theorem is unique.] measure μ on the locally compact space "X" and a unitary transformation "U" between "H"₀ and "L"²("X", μ) which carries the operators in $mathfrak{A}$ onto the corresponding multiplication operators.

The transformation "U" is called the spherical Fourier transform or sometimes just the spherical transform and μ is called the Plancherel measure. The Hilbert space "H"₀ can be identified with "L"²("K""G"/"K"), the space of "K"-biinvariant square integrable functions on "G".

The characters χ_λ of $mathfrak{A}$ (i.e. the points of "X") can be described by positive definite spherical functions φ_λ on "G", via the formula

: $chi_lambda(pi(f)) = int_G f(g)cdot varphi_lambda(g) , dg.$

for "f" in "C"_c("K""G"/"K"), where π("f") denotes the convolution operator in $mathfrak{A}$ and the integral is with respect to Haar measure on "G".

The spherical functions φ_λ on "G" are given by Harish-Chandra's formula:

In this formula:

* the integral is with respect to Haar measure on "K";
* λ is an element of "A"* =Hom("A",T) where "A" is the Abelian vector subgroup in the Iwasawa decomposition "G" ="KAN" of "G";
* λ' is defined on "G" by first extending λ to a character of the solvable subgroup "AN", using the group homomorphism onto "A", and then setting

:: $lambda^prime(kx) = Delta_{AN}(x)^{1/2} lambda(x)$

:for "k" in "K" and "x" in "AN", where Δ_"AN" is the modular function of "AN".

* Two different characters λ₁ and λ₂ give the same spherical function if and only if λ₁ = λ₂·"s", where "s" is in the Weyl group of "A"

:: $W=N_K(A)/C_K(A),$

:the quotient of the normaliser of "A" in "K" by its centraliser, a finite reflection group.

It follows that

* "X" can be identified with the quotient space "A"*/"W".

Harish-Chandra's c-function

Notes

References

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