# Plancherel theorem for spherical functions

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 X; it also gives the direct integral decomposition into irreducible representations of the regular representation on L2(X). In the case of hyperbolic space, these expansions were known from prior results of Mehler, Weyl and Fock.

The main reference for almost all this material is the encyclopedic text of Helgason (1984).

Contents 1 History 2 Spherical functions 3 Spherical principal series 4 Example: SL(2, C) 5 Example: SL(2, R) 5.1 Hadamard's method of descent 5.2 Flensted–Jensen's method of descent 5.3 Abel's integral equation 6 Other special cases 6.1 Complex semisimple Lie groups 6.2 Real semisimple Lie groups 7 Harish-Chandra's Plancherel theorem 8 Harish-Chandra's spherical function expansion 9 Harish-Chandra's c-function 10 Paley–Wiener theorem 11 Rosenberg's proof of inversion formula 12 Schwartz functions 13 Notes 14 References 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.[1] At around the same time, Harish-Chandra[2][3] and Gelfand & Naimark[4][5] 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 φλ 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[6][7] introduced his celebrated c-function c(λ) to describe the asymptotic behaviour of the spherical functions φλ and proposed c(λ)−2 dλ 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 Karpelevich to derive a product formula[8] 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.[9] In many special cases, for example for complex semisimple group or the Lorentz groups, there are simple methods to develop the theory directly. Certain subgroups of these groups can be treated by techniques generalising the well-known "method of descent" due to Jacques Hadamard. In particular Flensted-Jensen (1978) gave a general method for deducing properties of the spherical transform for a real semisimple group from that of its complexification.

One of the principal applications and motivations for the spherical transform was Selberg's trace formula. The classical Poisson summation formula combines the Fourier inversion formula on a vector group with summation over a cocompact lattice. In Selberg's analogue of this formula, the vector group is replaced by G/K, the Fourier transform by the spherical transform and the lattice by a cocompact (or cofinite) discrete subgroup. The original paper of Selberg (1956) implicitly invokes the spherical transform; it was Godement (1957) who brought the transform to the fore, giving in particular an elementary treatment for SL(2,R) along the lines sketched by Selberg.

Spherical functions Main article: Zonal spherical function Let G be a semisimple Lie group and K a maximal compact subgroup of G. The Hecke algebra Cc(K G/K), consisting of compactly supported K-biinvariant continuous functions on G, acts by convolution on the Hilbert space H=L2(G / K). Because G / K is a symmetric space, this *-algebra is commutative. The closure of its (the Hecke algebra's) image in the operator norm is a non-unital commutative C* algebra {displaystyle {mathfrak {A}}} , so by the Gelfand isomorphism can be identified with the continuous functions vanishing at infinity on its spectrum X.[10] Points in the spectrum are given by continuous *-homomorphisms of {displaystyle {mathfrak {A}}} into C, i.e. characters of {displaystyle {mathfrak {A}}} .

If S' denotes the commutant of a set of operators S on H, then {displaystyle {mathfrak {A}}^{prime }} can be identified with the commutant of the regular representation of G on H. Now {displaystyle {mathfrak {A}}} leaves invariant the subspace H0 of K-invariant vectors in H. Moreover, the abelian von Neumann algebra it generates on H0 is maximal Abelian. By spectral theory, there is an essentially unique[11] measure μ on the locally compact space X and a unitary transformation U between H0 and L2(X, μ) which carries the operators in {displaystyle {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 H0 can be identified with L2(KG/K), the space of K-biinvariant square integrable functions on G.

The characters χλ of {displaystyle {mathfrak {A}}} (i.e. the points of X) can be described by positive definite spherical functions φλ on G, via the formula {displaystyle chi _{lambda }(pi (f))=int _{G}f(g)cdot varphi _{lambda }(g),dg.} for f in Cc(KG/K), where π(f) denotes the convolution operator in {displaystyle {mathfrak {A}}} and the integral is with respect to Haar measure on G.

The spherical functions φλ on G are given by Harish-Chandra's formula: {displaystyle varphi _{lambda }(g)=int _{K}lambda ^{prime }(gk)^{-1},dk.} 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 {displaystyle lambda '(kx)=Delta _{AN}(x)^{1/2}lambda (x)} for k in K and x in AN, where ΔAN 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 {displaystyle 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. Spherical principal series See also: Principal series representation The spherical function φλ can be identified with the matrix coefficient of the spherical principal series of G. If M is the centralizer of A in K, this is defined as the unitary representation πλ of G induced by the character of B = MAN given by the composition of the homomorphism of MAN onto A and the character λ. The induced representation is defined on functions f on G with {displaystyle f(gb)=Delta (b)^{1/2}lambda (b)f(g)} for b in B by {displaystyle pi (g)f(x)=f(g^{-1}x),} where {displaystyle |f|^{2}=int _{K}|f(k)|^{2},dk0}} by Möbius transformations. The complex matrix {displaystyle g={begin{pmatrix}a&b\c&dend{pmatrix}}} acts as {displaystyle g(w)=(aw+b)(cw+d)^{-1}.} The stabiliser of the point j is the maximal compact subgroup K = SU(2), so that {displaystyle {mathfrak {H}}^{3}=G/K.} It carries the G-invariant Riemannian metric {displaystyle ds^{2}=r^{-2}left(dx^{2}+dy^{2}+dr^{2}right)} with associated volume element {displaystyle dV=r^{-3},dx,dy,dr} and Laplacian operator {displaystyle Delta =-r^{2}(partial _{x}^{2}+partial _{y}^{2}+partial _{r}^{2})+rpartial _{r}.} Every point in {displaystyle {mathfrak {H}}^{3}} can be written as k(etj) with k in SU(2) and t determined up to a sign. The Laplacian has the following form on functions invariant under SU(2), regarded as functions of the real parameter t: {displaystyle Delta =-partial _{t}^{2}-2coth tpartial _{t}.} The integral of an SU(2)-invariant function is given by {displaystyle int f,dV=int _{-infty }^{infty }f(t),sinh ^{2}t,dt.} Identifying the square integrable SU(2)-invariant functions with L2(R) by the unitary transformation Uf(t) = f(t) sinh t, Δ is transformed into the operator {displaystyle U^{*}Delta U=-{d^{2} over dt^{2}}+1.} By the Plancherel theorem and Fourier inversion formula for R, any SU(2)-invariant function f can be expressed in terms of the spherical functions {displaystyle Phi _{lambda }(t)={sin lambda t over lambda sinh t},} by the spherical transform {displaystyle {tilde {f}}(lambda )=int fPhi _{-lambda },dV} and the spherical inversion formula {displaystyle f(x)=int {tilde {f}}(lambda )Phi _{lambda }(x)lambda ^{2},dlambda .} Taking {displaystyle f=f_{2}^{*}star f_{1}} with fi in Cc(G / K) and {displaystyle f^{*}(g)={overline {f(g^{-1})}}} , and evaluating at i yields the Plancherel formula {displaystyle int _{G}f_{1}{overline {f_{2}}},dg=int {tilde {f}}_{1}(lambda ){overline {{tilde {f}}_{2}(lambda )}},lambda ^{2},dlambda .} For biinvariant functions this establishes the Plancherel theorem for spherical functions: the map {displaystyle {begin{cases}U:L^{2}(Kbackslash G/K)to L^{2}(mathbb {R} ,lambda ^{2},dlambda )\U:flongmapsto {tilde {f}}end{cases}}} is unitary and sends the convolution operator defined by {displaystyle fin L^{1}(Kbackslash G/K)} into the multiplication operator defined by {displaystyle {tilde {f}}} .

The spherical function Φλ is an eigenfunction of the Laplacian: {displaystyle Delta Phi _{lambda }=(lambda ^{2}+1)Phi _{lambda }.} Schwartz functions on R are the spherical transforms of functions f belonging to the Harish-Chandra Schwartz space {displaystyle {mathcal {S}}=left{fleft|sup _{t}left|(1+t^{2})^{N}(I+Delta )^{M}f(t)sinh(t)right|0}} by Möbius transformations. The real matrix {displaystyle g={begin{pmatrix}a&b\c&dend{pmatrix}}} acts as {displaystyle g(w)=(aw+b)(cw+d)^{-1}.} The stabiliser of the point i is the maximal compact subgroup K = SO(2), so that {displaystyle {mathfrak {H}}^{2}} = G / K. It carries the G-invariant Riemannian metric {displaystyle ds^{2}=r^{-2}left(dx^{2}+dr^{2}right)} with associated area element {displaystyle dA=r^{-2},dx,dr} and Laplacian operator {displaystyle Delta =-r^{2}(partial _{x}^{2}+partial _{r}^{2}).} Every point in {displaystyle {mathfrak {H}}^{2}} can be written as k( et i ) with k in SO(2) and t determined up to a sign. The Laplacian has the following form on functions invariant under SO(2), regarded as functions of the real parameter t: {displaystyle Delta =-partial _{t}^{2}-coth tpartial _{t}.} The integral of an SO(2)-invariant function is given by {displaystyle int f,dA=int _{-infty }^{infty }f(t)left|sinh tright|dt.} There are several methods for deriving the corresponding eigenfunction expansion for this ordinary differential equation including: the classical spectral theory of ordinary differential equations applied to the hypergeometric equation (Mehler, Weyl, Fock); variants of Hadamard's method of descent, realising 2-dimensional hyperbolic space as the quotient of 3-dimensional hyperbolic space by the free action of a 1-parameter subgroup of SL(2,C); Abel's integral equation, following Selberg and Godement; orbital integrals (Harish-Chandra, Gelfand & Naimark).

The second and third technique will be described below, with two different methods of descent: the classical one due Hadamard, familiar from treatments of the heat equation[12] and the wave equation[13] on hyperbolic space; and Flensted-Jensen's method on the hyperboloid.

Hadamard's method of descent If f(x,r) is a function on {displaystyle {mathfrak {H}}^{2}} and {displaystyle M_{1}f(x,y,r)=r^{1/2}cdot f(x,r)} then {displaystyle Delta _{3}M_{1}f=M_{1}left(Delta _{2}+{tfrac {3}{4}}right)f,} where Δn is the Laplacian on {displaystyle {mathfrak {H}}^{n}} .

Since the action of SL(2,C) commutes with Δ3, the operator M0 on S0(2)-invariant functions obtained by averaging M1f by the action of SU(2) also satisfies {displaystyle Delta _{3}M_{0}=M_{0}left(Delta _{2}+{tfrac {3}{4}}right).} The adjoint operator M1* defined by {displaystyle M_{1}^{*}F(x,r)=r^{1/2}int _{-infty }^{infty }F(x,y,r),dy} satisfies {displaystyle int _{{mathfrak {H}}^{3}}(M_{1}f)cdot F,dV=int _{{mathfrak {H}}^{2}}fcdot (M_{1}^{*}F),dA.} The adjoint M0*, defined by averaging M*f over SO(2), satisfies {displaystyle int _{{mathfrak {H}}^{3}}(M_{0}f)cdot F,dV=int _{{mathfrak {H}}^{2}}fcdot (M_{0}^{*}F),dA} for SU(2)-invariant functions F and SO(2)-invariant functions f. It follows that {displaystyle M_{i}^{*}Delta _{3}=left(Delta _{2}+{tfrac {3}{4}}right)M_{i}^{*}.} The function {displaystyle f_{lambda }=M_{1}^{*}Phi _{lambda }} is SO(2)-invariant and satisfies {displaystyle Delta _{2}f_{lambda }=left(lambda ^{2}+{tfrac {1}{4}}right)f_{lambda }.} On the other hand, {displaystyle b(lambda )=f_{lambda }(i)=int {sin lambda t over lambda sinh t},dt={pi over lambda }tanh {pi lambda over 2},} since the integral can be computed by integrating {displaystyle e^{ilambda t}/sinh t} around the rectangular indented contour with vertices at ±R and ±R + πi. Thus the eigenfunction {displaystyle phi _{lambda }=b(lambda )^{-1}M_{1}Phi _{lambda }} satisfies the normalisation condition φλ(i) = 1. There can only be one such solution either because the Wronskian of the ordinary differential equation must vanish or by expanding as a power series in sinh r.[14] It follows that {displaystyle varphi _{lambda }(e^{t}i)={frac {1}{2pi }}int _{0}^{2pi }left(cosh t-sinh tcos theta right)^{-1-ilambda },dtheta .} Similarly it follows that {displaystyle Phi _{lambda }=M_{1}phi _{lambda }.} If the spherical transform of an SO(2)-invariant function on {displaystyle {mathfrak {H}}^{2}} is defined by {displaystyle {tilde {f}}(lambda )=int fvarphi _{-lambda },dA,} then {displaystyle {(M_{1}^{*}F)}^{sim }(lambda )={tilde {F}}(lambda ).} Taking f=M1*F, the SL(2, C) inversion formula for F immediately yields {displaystyle f(x)=int _{-infty }^{infty }varphi _{lambda }(x){tilde {f}}(lambda ){lambda pi over 2}tanh left({pi lambda over 2}right),dlambda ,} the spherical inversion formula for SO(2)-invariant functions on {displaystyle {mathfrak {H}}^{2}} .

As for SL(2,C), this immediately implies the Plancherel formula for fi in Cc(SL(2,R) / SO(2)): {displaystyle int _{{mathfrak {H}}^{2}}f_{1}{overline {f_{2}}},dA=int _{-infty }^{infty }{tilde {f}}_{1}{overline {tilde {f_{2}}}}{lambda pi over 2}tanh left({pi lambda over 2}right),dlambda .} The spherical function φλ is an eigenfunction of the Laplacian: {displaystyle Delta _{2}varphi _{lambda }=left(lambda ^{2}+{tfrac {1}{4}}right)varphi _{lambda }.} Schwartz functions on R are the spherical transforms of functions f belonging to the Harish-Chandra Schwartz space {displaystyle {mathcal {S}}=left{fleft|sup _{t}left|(1+t^{2})^{N}(I+Delta )^{M}f(t)varphi _{0}(t)right|0}tanh {pi (alpha ,lambda ) over (alpha ,alpha )},} where α ranges over the positive roots of the root system in {displaystyle {mathfrak {a}}} and C is a normalising constant, given as a quotient of products of Gamma functions.

Harish-Chandra's Plancherel theorem Let G be a noncompact connected real semisimple Lie group with finite center. Let {displaystyle {mathfrak {g}}} denote its Lie algebra. Let K be a maximal compact subgroup given as the subgroup of fixed points of a Cartan involution σ. Let {displaystyle {mathfrak {g}}_{pm }} be the ±1 eigenspaces of σ in {displaystyle {mathfrak {g}}} , so that {displaystyle {mathfrak {k}}={mathfrak {g}}_{+}} is the Lie algebra of K and {displaystyle {mathfrak {p}}={mathfrak {g}}_{-}} give the Cartan decomposition {displaystyle {mathfrak {g}}={mathfrak {k}}+{mathfrak {p}},,,G=exp {mathfrak {p}}cdot K.} Let {displaystyle {mathfrak {a}}} be a maximal Abelian subalgebra of {displaystyle {mathfrak {p}}} and for α in {displaystyle {mathfrak {a}}^{*}} let {displaystyle {mathfrak {g}}_{alpha }={Xin {mathfrak {g}}:[H,X]=alpha (H)X,,(Hin {mathfrak {a}})}.} If α ≠ 0 and {displaystyle {mathfrak {g}}_{alpha }neq (0)} , then α is called a restricted root and {displaystyle m_{alpha }=dim {mathfrak {g}}_{alpha }} is called its multiplicity. Let A = exp {displaystyle {mathfrak {a}}} , so that G = KAK.The restriction of the Killing form defines an inner product on {displaystyle {mathfrak {p}}} and hence {displaystyle {mathfrak {a}}} , which allows {displaystyle {mathfrak {a}}^{*}} to be identified with {displaystyle {mathfrak {a}}} . With respect to this inner product, the restricted roots Σ give a root system. Its Weyl group can be identified with {displaystyle W=N_{K}(A)/C_{K}(A)} . A choice of positive roots defines a Weyl chamber {displaystyle {mathfrak {a}}_{+}^{*}} . The reduced root system Σ0 consists of roots α such that α/2 is not a root.

Defining the spherical functions φ λ as above for λ in {displaystyle {mathfrak {a}}^{*}} , the spherical transform of f in Cc∞(K G / K) is defined by {displaystyle {tilde {f}}(lambda )=int _{G}f(g)varphi _{-lambda }(g),dg.} The spherical inversion formula states that {displaystyle f(g)=int _{{mathfrak {a}}_{+}^{*}}{tilde {f}}(lambda )varphi _{lambda }(g),|c(lambda )|^{-2},dlambda ,} where Harish-Chandra's c-function c(λ) is defined by[33] {displaystyle c(lambda )=c_{0}cdot prod _{alpha in Sigma _{0}^{+}}{frac {2^{-i(lambda ,alpha _{0})}Gamma (i(lambda ,alpha _{0}))}{Gamma !left({frac {1}{2}}left[{frac {1}{2}}m_{alpha }+1+i(lambda ,alpha _{0})right]right)Gamma !left({frac {1}{2}}left[{frac {1}{2}}m_{alpha }+m_{2alpha }+i(lambda ,alpha _{0})right]right)}}} with {displaystyle alpha _{0}=(alpha ,alpha )^{-1}alpha } and the constant c0 chosen so that c(−iρ) = 1 where {displaystyle rho ={frac {1}{2}}sum _{alpha in Sigma ^{+}}m_{alpha }alpha .} The Plancherel theorem for spherical functions states that the map {displaystyle W:fmapsto {tilde {f}},,,, L^{2}(Kbackslash G/K)rightarrow L^{2}({mathfrak {a}}_{+}^{*},|c(lambda )|^{-2},dlambda )} is unitary and transforms convolution by {displaystyle fin L^{1}(Kbackslash G/K)} into multiplication by {displaystyle {tilde {f}}} .

Harish-Chandra's spherical function expansion Since G = KAK, functions on G/K that are invariant under K can be identified with functions on A, and hence {displaystyle {mathfrak {a}}} , that are invariant under the Weyl group W. In particular since the Laplacian Δ on G/K commutes with the action of G, it defines a second order differential operator L on {displaystyle {mathfrak {a}}} , invariant under W, called the radial part of the Laplacian. In general if X is in {displaystyle {mathfrak {a}}} , it defines a first order differential operator (or vector field) by {displaystyle Xf(y)=left.{frac {d}{dt}}f(y+tX)right|_{t=0}.} L can be expressed in terms of these operators by the formula[34] {displaystyle L=Delta _{mathfrak {a}}-sum _{alpha >0}m_{alpha },coth alpha ,A_{alpha },} where Aα in {displaystyle {mathfrak {a}}} is defined by {displaystyle (A_{alpha },X)=alpha (X)} and {displaystyle Delta _{mathfrak {a}}=-sum X_{i}^{2}} is the Laplacian on {displaystyle {mathfrak {a}}} , corresponding to any choice of orthonormal basis (Xi).

Thus {displaystyle L=L_{0}-sum _{alpha >0}m_{alpha },(coth alpha -1)A_{alpha },} where {displaystyle L_{0}=Delta _{mathfrak {a}}-sum _{alpha >0}A_{alpha },} so that L can be regarded as a perturbation of the constant-coefficient operator L0.

Now the spherical function φλ is an eigenfunction of the Laplacian: {displaystyle Delta varphi _{lambda }=left(left|lambda right|^{2}+left|rho right|^{2}right)varphi _{lambda }} and therefore of L, when viewed as a W-invariant function on {displaystyle {mathfrak {a}}} .

Since eiλ–ρ and its transforms under W are eigenfunctions of L0 with the same eigenvalue, it is natural look for a formula for φλ in terms of a perturbation series {displaystyle f_{lambda }=e^{ilambda -rho }sum _{mu in Lambda }a_{mu }(lambda )e^{-mu },} with Λ the cone of all non-negative integer combinations of positive roots, and the transforms of fλ under W. The expansion {displaystyle coth x-1=2sum _{m>0}e^{-2mx},} leads to a recursive formula for the coefficients aμ(λ). In particular they are uniquely determined and the series and its derivatives converges absolutely on {displaystyle {mathfrak {a}}_{+}} , a fundamental domain for W. Remarkably it turns out that fλ is also an eigenfunction of the other G-invariant differential operators on G/K, each of which induces a W-invariant differential operator on {displaystyle {mathfrak {a}}} .

It follows that φλ can be expressed in terms as a linear combination of fλ and its transforms under W:[35] {displaystyle varphi _{lambda }=sum _{sin W}c(slambda )f_{slambda }.} Here c(λ) is Harish-Chandra's c-function. It describes the asymptotic behaviour of φλ in {displaystyle {mathfrak {a}}_{+}} , since[36] {displaystyle varphi _{lambda }(e^{t}X)sim c(lambda )e^{(ilambda -rho )Xt}} for X in {displaystyle {mathfrak {a}}_{+}} and t > 0 large.

Harish-Chandra obtained a second integral formula for φλ and hence c(λ) using the Bruhat decomposition of G:[37] {displaystyle G=bigcup _{sin W}BsB,} where B = MAN and the union is disjoint. Taking the Coxeter element s0 of W, the unique element mapping {displaystyle {mathfrak {a}}_{+}} onto {displaystyle -{mathfrak {a}}_{+}} , it follows that σ(N) has a dense open orbit G/B = K/M whose complement is a union of cells of strictly smaller dimension and therefore has measure zero. It follows that the integral formula for φλ initially defined over K/M {displaystyle varphi _{lambda }(g)=int _{K/M}lambda '(gk)^{-1},dk.} can be transferred to σ(N):[38] {displaystyle varphi _{lambda }(e^{X})=e^{ilambda -rho }int _{sigma (N)}{{overline {lambda '(n)}} over lambda '(e^{X}ne^{-X})},dn,} for X in {displaystyle {mathfrak {a}}} .

Since {displaystyle lim _{tto infty }e^{tX}ne^{-tX}=1} for X in {displaystyle {mathfrak {a}}_{+}} , the asymptotic behaviour of φλ can be read off from this integral, leading to the formula:[39] {displaystyle c(lambda )=int _{sigma (N)}{overline {lambda '(n)}},dn.} Harish-Chandra's c-function Main article: Harish-Chandra's c-function The many roles of Harish-Chandra's c-function in non-commutative harmonic analysis are surveyed in Helgason (2000). Although it was originally introduced by Harish-Chandra in the asymptotic expansions of spherical functions, discussed above, it was also soon understood to be intimately related to intertwining operators between induced representations, first studied in this context by Bruhat (1957). These operators exhibit the unitary equivalence between πλ and πsλ for s in the Weyl group and a c-function cs(λ) can be attached to each such operator: namely the value at 1 of the intertwining operator applied to ξ0, the constant function 1, in L2(K/M).[40] Equivalently, since ξ0 is up to scalar multiplication the unique vector fixed by K, it is an eigenvector of the intertwining operator with eigenvalue cs(λ). These operators all act on the same space L2(K/M), which can be identified with the representation induced from the 1-dimensional representation defined by λ on MAN. Once A has been chosen, the compact subgroup M is uniquely determined as the centraliser of A in K. The nilpotent subgroup N, however, depends on a choice of a Weyl chamber in {displaystyle {mathfrak {a}}^{*}} , the various choices being permuted by the Weyl group W = M ' / M, where M ' is the normaliser of A in K. The standard intertwining operator corresponding to (s, λ) is defined on the induced representation by[41] {displaystyle A(s,lambda )F(k)=int _{sigma (N)cap s^{-1}Ns}F(ksn),dn,} where σ is the Cartan involution. It satisfies the intertwining relation {displaystyle A(s,lambda )pi _{lambda }(g)=pi _{slambda }(g)A(s,lambda ).} The key property of the intertwining operators and their integrals is the multiplicative cocycle property[42] {displaystyle A(s_{1}s_{2},lambda )=A(s_{1},s_{2}lambda )A(s_{2},lambda ),} whenever {displaystyle ell (s_{1}s_{2})=ell (s_{1})+ell (s_{2})} for the length function on the Weyl group associated with the choice of Weyl chamber. For s in W, this is the number of chambers crossed by the straight line segment between X and sX for any point X in the interior of the chamber. The unique element of greatest length s0, namely the number of positive restricted roots, is the unique element that carries the Weyl chamber {displaystyle {mathfrak {a}}_{+}^{*}} onto {displaystyle -{mathfrak {a}}_{+}^{*}} . By Harish-Chandra's integral formula, it corresponds to Harish-Chandra's c-function: {displaystyle c(lambda )=c_{s_{0}}(lambda ).} The c-functions are in general defined by the equation {displaystyle A(s,lambda )xi _{0}=c_{s}(lambda )xi _{0},} where ξ0 is the constant function 1 in L2(K/M). The cocycle property of the intertwining operators implies a similar multiplicative property for the c-functions: {displaystyle c_{s_{1}s_{2}}(lambda )=c_{s_{1}}(s_{2}lambda )c_{s_{2}}(lambda )} provided {displaystyle ell (s_{1}s_{2})=ell (s_{1})+ell (s_{2}).} This reduces the computation of cs to the case when s = sα, the reflection in a (simple) root α, the so-called "rank-one reduction" of Gindikin & Karpelevich (1962). In fact the integral involves only the closed connected subgroup Gα corresponding to the Lie subalgebra generated by {displaystyle {mathfrak {g}}_{pm alpha }} where α lies in Σ0+.[43] Then Gα is a real semisimple Lie group with real rank one, i.e. dim Aα = 1, and cs is just the Harish-Chandra c-function of Gα. In this case the c-function can be computed directly by various means: by noting that φλ can be expressed in terms of the hypergeometric function for which the asymptotic expansion is known from the classical formulas of Gauss for the connection coefficients;[6][44] by directly computing the integral, which can be expressed as an integral in two variables and hence a product of two beta functions.[45][46] This yields the following formula: {displaystyle c_{s_{alpha }}(lambda )=c_{0}{frac {2^{-i(lambda ,alpha _{0})}Gamma (i(lambda ,alpha _{0}))}{Gamma !left({frac {1}{2}}left({frac {1}{2}}m_{alpha }+1+i(lambda ,alpha _{0})right)right)Gamma !left({frac {1}{2}}left({frac {1}{2}}m_{alpha }+m_{2alpha }+i(lambda ,alpha _{0})right)right)}},} where {displaystyle c_{0}=2^{m_{alpha }/2+m_{2alpha }}Gamma !left({tfrac {1}{2}}(m_{alpha }+m_{2alpha }+1)right).} The general Gindikin–Karpelevich formula for c(λ) is an immediate consequence of this formula and the multiplicative properties of cs(λ).

Paley–Wiener theorem The Paley-Wiener theorem generalizes the classical Paley-Wiener theorem by characterizing the spherical transforms of smooth K-bivariant functions of compact support on G. It is a necessary and sufficient condition that the spherical transform be W-invariant and that there is an R > 0 such that for each N there is an estimate {displaystyle |{tilde {f}}(lambda )|leq C_{N}(1+|lambda |)^{-N}e^{Rleft|operatorname {Im} lambda right|}.} In this case f is supported in the closed ball of radius R about the origin in G/K.

This was proved by Helgason and Gangolli (Helgason (1970) pg. 37).

The theorem was later proved by Flensted-Jensen (1986) independently of the spherical inversion theorem, using a modification of his method of reduction to the complex case.[47] Rosenberg's proof of inversion formula Rosenberg (1977) noticed that the Paley-Wiener theorem and the spherical inversion theorem could be proved simultaneously, by a trick which considerably simplified previous proofs.

The first step of his proof consists in showing directly that the inverse transform, defined using Harish-Chandra's c-function, defines a function supported in the closed ball of radius R about the origin if the Paley-Wiener estimate is satisfied. This follows because the integrand defining the inverse transform extends to a meromorphic function on the complexification of {displaystyle {mathfrak {a}}^{*}} ; the integral can be shifted to {displaystyle {mathfrak {a}}^{*}+imu t} for μ in {displaystyle {mathfrak {a}}_{+}^{*}} and t > 0. Using Harish-Chandra's expansion of φλ and the formulas for c(λ) in terms of Gamma functions, the integral can be bounded for t large and hence can be shown to vanish outside the closed ball of radius R about the origin.[48] This part of the Paley-Wiener theorem shows that {displaystyle T(f)=int _{{mathfrak {a}}_{+}^{*}}{tilde {f}}(lambda )|c(lambda )|^{-2},dlambda } defines a distribution on G/K with support at the origin o. A further estimate for the integral shows that it is in fact given by a measure and that therefore there is a constant C such that {displaystyle T(f)=Cf(o).} By applying this result to {displaystyle f_{1}(g)=int _{K}f(x^{-1}kg),dk,} it follows that {displaystyle Cf=int _{{mathfrak {a}}_{+}^{*}}{tilde {f}}(lambda )varphi _{lambda }|c(lambda )|^{-2},dlambda .} A further scaling argument allows the inequality C = 1 to be deduced from the Plancherel theorem and Paley-Wiener theorem on {displaystyle {mathfrak {a}}} .[49][50] Schwartz functions The Harish-Chandra Schwartz space can be defined as[51] {displaystyle {mathcal {S}}(Kbackslash G/K)=left{fin C^{infty }(G/K)^{K}:sup _{x}left|(1+d(x,o))^{m}(Delta +I)^{n}f(x)right|

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