- Plancherel theorem for spherical functions
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 analysisof the Plancherel formulaand Fourier inversion formulain the representation theory of the group of real numbers in classical harmonic analysisand has a similarly close interconnection with the theory of differential equations.It is the special case for zonal spherical functions of the general Plancherel theoremfor semisimple Lie groups, also proved by Harish-Chandra. The Plancherel theorem gives the eigenfunction expansion of radial functions for the Laplacian operatoron the associated symmetric space. In the case of hyperbolic space, these expansions were known from prior results of Mehler, Weyland Fock.
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 groupthese spherical functions φλ were naturally labelled by a parameter λ in the quotient of a Euclidean spaceby the action of a finite reflection group, it became a central problem to determine explicitly the Plancherel measurein terms of this parametrization. Generalizing the ideas of Hermann Weylfrom 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 φλ 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]
Let "G" be a semisimple
Lie groupand "K" a maximal compact subgroupof "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, so by the Gelfand isomorphismcan 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 .] Points in the spectrum are given by continuous *-homomorphisms of into C, i.e. characters of .
If "S"' denotes the
commutantof a set of operators "S" on "H", then can be identified with the commutant of the regular representationof "G" on "H". Now leaves invariant the subspace "H"0 of "K"-invariant vectors in "H". Moreover the Abelian von Neumann algebrait 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 theoremis unique.] measureμ on the locally compactspace "X" and a unitary transformation "U" between "H"0 and "L"2("X", μ) which carries the operators in 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".
for "f" in "C"c("K""G"/"K"), where π("f") denotes the convolution operator in and the integral is with respect to
Haar measureon "G".
The spherical functions φλ 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
: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 groupof "A"
:the quotient of the
normaliserof "A" in "K" by its centraliser, a finite reflection group.
It follows that
* "X" can be identified with the quotient space "A"*/"W".
*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
Wikimedia Foundation. 2010.
Look at other dictionaries:
Commutation theorem — In mathematics, a commutation theorem explicitly identifies the commutant of a specific von Neumann algebra acting on a Hilbert space in the presence of a trace. The first such result was proved by F.J. Murray and John von Neumann in the 1930s… … Wikipedia
Michel Plancherel — (16 January 1885, Bussy, Fribourg 4 March 1967, Zurich) was a Swiss mathematician. He was born in Bussy (Fribourg, Switzerland) and obtained his diploma in mathematics from the University of Fribourg in 1907. He was a professor in Fribourg (1911) … Wikipedia
Zonal spherical function — In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group G with compact subgroup K (often a maximal compact subgroup) that arises as the matrix coefficient of a K invariant vector in an… … Wikipedia
Hilbert space — For the Hilbert space filling curve, see Hilbert curve. Hilbert spaces can be used to study the harmonics of vibrating strings. The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It… … Wikipedia
List of mathematics articles (P) — NOTOC P P = NP problem P adic analysis P adic number P adic order P compact group P group P² irreducible P Laplacian P matrix P rep P value P vector P y method Pacific Journal of Mathematics Package merge algorithm Packed storage matrix Packing… … Wikipedia
List of theorems — This is a list of theorems, by Wikipedia page. See also *list of fundamental theorems *list of lemmas *list of conjectures *list of inequalities *list of mathematical proofs *list of misnamed theorems *Existence theorem *Classification of finite… … Wikipedia
Spectral theory of ordinary differential equations — In mathematics, the spectral theory of ordinary differential equations is concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his dissertation Hermann Weyl… … Wikipedia
Tempered representation — In mathematics, a tempered representation of a linear semisimple Lie group is a representation that has a basis whose matrix coefficients lie in the L p space : L 2+ epsilon;( G ) for any epsilon; gt; 0. FormulationThis condition, as just given,… … Wikipedia
Fourier transform — Fourier transforms Continuous Fourier transform Fourier series Discrete Fourier transform Discrete time Fourier transform Related transforms The Fourier transform is a mathematical operation that decomposes a function into its constituent… … Wikipedia
Fourier series — Fourier transforms Continuous Fourier transform Fourier series Discrete Fourier transform Discrete time Fourier transform Related transforms … Wikipedia