- Plancherel theorem for spherical functions
In

mathematics , the**Plancherel theorem for spherical functions**is an important result in the representation theory ofsemisimple Lie group s, due in its final form toHarish-Chandra . It is a natural generalisation innon-commutative harmonic analysis of thePlancherel formula andFourier inversion formula in the representation theory of the group of real numbers in classicalharmonic analysis and has a similarly close interconnection with the theory ofdifferential equation s.It is the special case forzonal spherical function s of the generalPlancherel theorem for semisimple Lie groups, also proved by Harish-Chandra. The Plancherel theorem gives the eigenfunction expansion of radial functions for theLaplacian operator on the associatedsymmetric space . In the case ofhyperbolic space , these expansions were known from prior results of Mehler,Weyl andFock .**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 forSL(2,R) and complexsemisimple Lie group s, so in particular theLorentz group s. A simpler abstract formula was derived by Mautner for a "topological" symmetric space "G"/"K" corresponding to amaximal compact subgroup "K". Godement gave a more concrete and satisfactory form for positive definite spherical functions, a class ofspecial function s on "G"/"K". Since when "G" is asemisimple Lie group these spherical functions φ_{λ}were naturally labelled by a parameter λ in the quotient of aEuclidean space by the action of afinite reflection group , it became a central problem to determine explicitly thePlancherel measure in terms of this parametrization. Generalizing the ideas ofHermann Weyl from thespectral 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 ahyperbolic 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" amaximal compact subgroup of "G". TheHecke algebra "C"_{c}("K" "G"/"K"), consisting of compactly supported "K"-biinvariant continuous functions on "G", acts by convolution on theHilbert space "H"="L"^{2}("G" / "K"). Because "G" / "K" is asymmetric space , this *-algebra iscommutative . The closure of its image in the operator norm is a non-unital commutativeC* algebra $mathfrak\{A\}$, so by theGelfand isomorphism can be identified with the continuous functions vanishing at infinity on itsspectrum "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 theregular representation of "G" on "H". Now $mathfrak\{A\}$ leaves invariant the subspace "H"_{0}of "K"-invariant vectors in "H". Moreover theAbelian von Neumann algebra it generates on "H"_{0}is maximal Abelian. Byspectral theory , there is an essentially unique [*The measure class of μ in the sense of the*]Radon-Nikodym theorem is unique.measure μ on thelocally 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 correspondingmultiplication operator s.The transformation "U" is called the

**spherical Fourier transform**or sometimes just the**spherical transform**and μ is called the. The Hilbert space "H"Plancherel measure _{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 toHaar 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 theIwasawa 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 λ

_{1}and λ_{2}give the same spherical function if and only if λ_{1}= λ_{2}·"s", where "s" is in theWeyl group of "A"::$W=N\_K(A)/C\_K(A),$

:the quotient of the

normaliser of "A" in "K" by itscentraliser , afinite reflection group .It follows that

*

**"X" can be identified with the quotient space "A"*/"W**".**Harish-Chandra's c-function****Notes****References***citation|first=Jean|last=Dieudonné|authorlink=Jean Dieudonné|title=Treatise on Analysis, Vol. VI|publisher=Academic Press|year=1978|id =ISBN 0-12-215506-8

*citation|first=I.M.|last=Gelfand|authorlink=Israel Gelfand|first2=M.A.|last2=Naimark|authorlink2=Mark Naimark|year=1948| title= An analog of Plancherel's formula for the complex unimodular group|journal=Dokl. Akad. Nauk USSR|volume=63|pages=609-612

*citation|first=S.G.|last=Gindikin|first2=F.I.|last2=Karpelevič|title=Plancherel measure of Riemannian symmetric spaces of non-positive curvature|year=1962|volume=145|journal=Dokl. Akad. Nauk. SSSR|pages=252-254

*citation|first=Victor|last=Guillemin|authorlink=Victor Guillemin|first2=Shlomo|last2=Sternberg|authorlink2=Shlomo Sternberg|title=Geometric Asymptotics|publisher=American Mathematical Society|year=1977

id=ISBN 0821816330, Appendix to Chapter VI, "The Plancherel Formula for Complex Semisimple Lie Groups".

*citation|last=Harish-Chandra|authorlink=Harish-Chandra|title=Plancherel formula for complex semisimple Lie groups|journal=Proc. Nat. Acad. Sci. U.S.A.|year=1951|volume=37|pages=813-818|url=http://www.jstor.org/stable/88521

*citation|last=Harish-Chandra|authorlink=Harish-Chandra|title=Plancherel formula for the 2 x 2 real unimodular group|journal=Proc. Nat. Acad. Sci. U.S.A.|year=1952|volume=38|pages=337-342|url=http://www.jstor.org/stable/88737

*Citation | last1=Harish-Chandra |authorlink=Harish-Chandra| title=Spherical functions on a semisimple Lie group. I | url=http://www.jstor.org/stable/2372786 | id=MathSciNet | id = 0094407 | year=1958a | journal=American Journal of Mathematics | issn=0002-9327 | volume=80 | pages=241–310

*Citation | last1=Harish-Chandra |authorlink=Harish-Chandra| title=Spherical Functions on a Semisimple Lie Group II | url=http://www.jstor.org/stable/2372772 | publisher=The Johns Hopkins University Press | year=1958b | journal=American Journal of Mathematics | issn=0002-9327 | volume=80 | issue=3 | pages=553–613

*citation|last=Harish-Chandra|authorlink=Harish-Chandra|year=1966|title=Discrete series for semisimple Lie groups, II.|journal=Acta Mathematica|volume=116|pages=1-111|doi=10.1007/BF02392813, section 21.

*citation|first=Sigurdur|last=Helgason|title=Groups and Geometric Analysis. Integral Geometry, Invariant Differential Operators and Spherical Functions|publisher=Academic Press|year=1984|id=ISBN 0-12-338301-3

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