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).

Conteúdo 1 História 2 Spherical functions 3 Spherical principal series 4 Exemplo: SL(2, C) 5 Exemplo: 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 Notas 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 caso. 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 {estilo de exibição {mathfrak {UMA}}} , 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 {estilo de exibição {mathfrak {UMA}}} into C, ou seja. characters of {estilo de exibição {mathfrak {UMA}}} .

If S' denotes the commutant of a set of operators S on H, então {estilo de exibição {mathfrak {UMA}}^{melhor }} can be identified with the commutant of the regular representation of G on H. Agora {estilo de exibição {mathfrak {UMA}}} leaves invariant the subspace H0 of K-invariant vectors in H. Além disso, 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, m) which carries the operators in {estilo de exibição {mathfrak {UMA}}} 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 {estilo de exibição {mathfrak {UMA}}} (ou seja. the points of X) can be described by positive definite spherical functions φλ on G, via the formula {estilo de exibição chi _{lambda }(pi (f))=int_{G}f(g)cdot varphi _{lambda }(g),dg.} for f in Cc(KG/K), onde π(f) denotes the convolution operator in {estilo de exibição {mathfrak {UMA}}} and the integral is with respect to Haar measure on G.

The spherical functions φλ on G are given by Harish-Chandra's formula: {estilo de exibição varphi _{lambda }(g)=int_{K}lambda ^{melhor }(gk)^{-1},dk.} In this formula: the integral is with respect to Haar measure on K; λ is an element of A* =Hom(UMA,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}(UMA)/C_{K}(UMA),} 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 {estilo de exibição f(gb)=Delta (b)^{1/2}lambda (b)f(g)} for b in B by {estilo de exibição pi (g)f(x)=f(g^{-1}x),} Onde {estilo de exibição |f|^{2}=int_{K}|f(k)|^{2},dk

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 {estilo de exibição {matemática {S}}= esquerda{abandonou|e aí _{t}deixei|(1+t^{2})^{N}(I+Delta )^{M}f(t)sinh(t)certo|0}} Möbius transformations. real {displaystyle g={begin{pmatrix}a&b\c&dend{pmatrix}}} {displaystyle g(w)=(aw+b)(cw+d)^{-1}.} stabiliser point i maximal subgroup K =SO(2), so that {displaystyle {mathfrak {H}}^{2}} =G >

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 {estilo de exibição {mathfrak {H}}^{2}} e {estilo de exibição M_{1}f(x,y,r)=r^{1/2}cdot f(x,r)} então {estilo de exibição Delta _{3}M_{1}f=M_{1}deixei(Delta_{2}+{tfrac {3}{4}}certo)f,} where Δn is the Laplacian on {estilo de exibição {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 {estilo de exibição Delta _{3}M_{0}=M_{0}deixei(Delta_{2}+{tfrac {3}{4}}certo).} The adjoint operator M1* defined by {estilo de exibição M_{1}^{*}F(x,r)=r^{1/2}int_{-infty }^{infty }F(x,y,r),dy} satisfies {estilo de exibição 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 {estilo de exibição 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. Segue que {estilo de exibição M_{eu}^{*}Delta_{3}= esquerda(Delta_{2}+{tfrac {3}{4}}certo)M_{eu}^{*}.} A função {estilo de exibição f_{lambda }=M_{1}^{*}Phi _{lambda }} is SO(2)-invariant and satisfies {estilo de exibição Delta _{2}f_{lambda }= esquerda(lambda ^{2}+{tfrac {1}{4}}certo)f_{lambda }.} Por outro lado, {estilo de exibição b(lambda )=f_{lambda }(eu)=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 {estilo de exibição e^{ilambda t}/sinh t} around the rectangular indented contour with vertices at ±R and ±R + πi. Thus the eigenfunction {estilo de exibição phi _{lambda }=b(lambda )^{-1}M_{1}Phi _{lambda }} satisfies the normalisation condition φλ(eu) = 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] Segue que {estilo de exibição varphi _{lambda }(e^{t}eu)={fratura {1}{2pi }}int_{0}^{2pi }deixei(cosh t-sinh tcos theta right)^{-1-ilambda },teta .} Similarly it follows that {displaystyle Phi _{lambda }=M_{1}phi_{lambda }.} If the spherical transform of an SO(2)-invariant function on {estilo de exibição {mathfrak {H}}^{2}} é definido por {estilo de exibição {tilde {f}}(lambda )=int fvarphi _{-lambda },dA,} então {estilo de exibição {(M_{1}^{*}F)}^{sim }(lambda )={tilde {F}}(lambda ).} Taking f=M1*F, the SL(2, C) inversion formula for F immediately yields {estilo de exibição f(x)=int_{-infty }^{infty }varphi_{lambda }(x){tilde {f}}(lambda ){lambda pi over 2}tanh left({pi lambda over 2}certo),dlambda ,} the spherical inversion formula for SO(2)-invariant functions on {estilo de exibição {mathfrak {H}}^{2}} .

As for SL(2,C), this immediately implies the Plancherel formula for fi in Cc(SL(2,R) / SO(2)): {estilo de exibição 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}certo),dlambda .} The spherical function φλ is an eigenfunction of the Laplacian: {estilo de exibição Delta _{2}varphi_{lambda }= esquerda(lambda ^{2}+{tfrac {1}{4}}certo)varphi_{lambda }.} Schwartz functions on R are the spherical transforms of functions f belonging to the Harish-Chandra Schwartz space {estilo de exibição {matemática {S}}= esquerda{abandonou|e aí _{t}deixei|(1+t^{2})^{N}(I+Delta )^{M}f(t)varphi_{0}(t)certo|

Harish-Chandra's Plancherel theorem Let G be a noncompact connected real semisimple Lie group with finite center. Deixar {estilo de exibição {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. Deixar {estilo de exibição {mathfrak {g}}_{PM }} be the ±1 eigenspaces of σ in {estilo de exibição {mathfrak {g}}} , de modo a {estilo de exibição {mathfrak {k}}={mathfrak {g}}_{+}} is the Lie algebra of K and {estilo de exibição {mathfrak {p}}={mathfrak {g}}_{-}} give the Cartan decomposition {estilo de exibição {mathfrak {g}}={mathfrak {k}}+{mathfrak {p}},,,G=exp {mathfrak {p}}cdot K.} Deixar {estilo de exibição {mathfrak {uma}}} be a maximal Abelian subalgebra of {estilo de exibição {mathfrak {p}}} and for α in {estilo de exibição {mathfrak {uma}}^{*}} deixar {estilo de exibição {mathfrak {g}}_{alfa }={Xin {mathfrak {g}}:[H,X]= alfa (H)X,,(Hin {mathfrak {uma}})}.} If α ≠ 0 e {estilo de exibição {mathfrak {g}}_{alfa }neq (0)} , then α is called a restricted root and {estilo de exibição m_{alfa }=dim {mathfrak {g}}_{alfa }} is called its multiplicity. Let A = exp {estilo de exibição {mathfrak {uma}}} , so that G = KAK.The restriction of the Killing form defines an inner product on {estilo de exibição {mathfrak {p}}} e, portanto {estilo de exibição {mathfrak {uma}}} , which allows {estilo de exibição {mathfrak {uma}}^{*}} to be identified with {estilo de exibição {mathfrak {uma}}} . 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}(UMA)/C_{K}(UMA)} . A choice of positive roots defines a Weyl chamber {estilo de exibição {mathfrak {uma}}_{+}^{*}} . 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 {estilo de exibição {mathfrak {uma}}^{*}} , the spherical transform of f in Cc∞(K G / K) é definido por {estilo de exibição {tilde {f}}(lambda )=int_{G}f(g)varphi_{-lambda }(g),dg.} The spherical inversion formula states that {estilo de exibição f(g)=int_{{mathfrak {uma}}_{+}^{*}}{tilde {f}}(lambda )varphi_{lambda }(g),|c(lambda )|^{-2},dlambda ,} where Harish-Chandra's c-function c(λ) é definido por[33] {estilo de exibição c(lambda )=c_{0}cdot prod _{alpha in Sigma _{0}^{+}}{fratura {2^{-eu(lambda ,alfa _{0})}Gama (eu(lambda ,alfa _{0}))}{Gama !deixei({fratura {1}{2}}deixei[{fratura {1}{2}}m_{alfa }+1+eu(lambda ,alfa _{0})certo]certo)Gama !deixei({fratura {1}{2}}deixei[{fratura {1}{2}}m_{alfa }+m_{2alfa }+eu(lambda ,alfa _{0})certo]certo)}}} com {alfa de estilo de exibição _{0}=(alfa ,alfa )^{-1}alfa } and the constant c0 chosen so that c(−iρ) = 1 Onde {displaystyle rho ={fratura {1}{2}}soma _{alpha in Sigma ^{+}}m_{alfa }alfa .} The Plancherel theorem for spherical functions states that the map {estilo de exibição W.:fmapsto {tilde {f}},,,, L^{2}(Kbackslash G/K)rightarrow L^{2}({mathfrak {uma}}_{+}^{*},|c(lambda )|^{-2},dlambda )} is unitary and transforms convolution by {displaystyle fin L^{1}(Kbackslash G/K)} into multiplication by {estilo de exibição {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, e, portanto {estilo de exibição {mathfrak {uma}}} , 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 {estilo de exibição {mathfrak {uma}}} , invariant under W, called the radial part of the Laplacian. In general if X is in {estilo de exibição {mathfrak {uma}}} , it defines a first order differential operator (or vector field) por {displaystyle Xf(y)=left.{fratura {d}{dt}}f(y+tX)certo|_{t=0}.} L can be expressed in terms of these operators by the formula[34] {displaystyle L=Delta _{mathfrak {uma}}-soma _{alpha >0}m_{alfa },coth alpha ,UMA_{alfa },} where Aα in {estilo de exibição {mathfrak {uma}}} é definido por {estilo de exibição (UMA_{alfa },X)= alfa (X)} e {estilo de exibição Delta _{mathfrak {uma}}=-sum X_{eu}^{2}} is the Laplacian on {estilo de exibição {mathfrak {uma}}} , corresponding to any choice of orthonormal basis (XI).

Desta forma {displaystyle L=L_{0}-soma _{alpha >0}m_{alfa },(coth alpha -1)UMA_{alfa },} Onde {estilo de exibição L_{0}=Delta _{mathfrak {uma}}-soma _{alpha >0}UMA_{alfa },} 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 }= esquerda(deixei|lambda certo|^{2}+deixei|rho right|^{2}certo)varphi_{lambda }} and therefore of L, when viewed as a W-invariant function on {estilo de exibição {mathfrak {uma}}} .

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 {estilo de exibição f_{lambda }=e^{ilambda -rho }soma _{mu in Lambda }uma_{dentro }(lambda )e^{-dentro },} 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 {estilo de exibição {mathfrak {uma}}_{+}} , 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 {estilo de exibição {mathfrak {uma}}} .

It follows that φλ can be expressed in terms as a linear combination of fλ and its transforms under W:[35] {estilo de exibição varphi _{lambda }=soma _{sin W}c(slambda )f_{slambda }.} Here c(λ) is Harish-Chandra's c-function. It describes the asymptotic behaviour of φλ in {estilo de exibição {mathfrak {uma}}_{+}} , desde[36] {estilo de exibição varphi _{lambda }(e^{t}X)sim c(lambda )e^{(ilambda -rho )Xt}} for X in {estilo de exibição {mathfrak {uma}}_{+}} 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 {estilo de exibição {mathfrak {uma}}_{+}} onto {estilo de exibição -{mathfrak {uma}}_{+}} , 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 {estilo de exibição varphi _{lambda }(g)=int_{K/M}lambda '(gk)^{-1},dk.} can be transferred to σ(N):[38] {estilo de exibição varphi _{lambda }(e^{X})=e^{ilambda -rho }int_{sigma (N)}{{overline {lambda '(n)}} over lambda '(e^{X}ne^{-X})},dn,} for X in {estilo de exibição {mathfrak {uma}}} .

Desde {displaystyle lim _{tto infty }e^{tX}ne^{-tX}=1} for X in {estilo de exibição {mathfrak {uma}}_{+}} , the asymptotic behaviour of φλ can be read off from this integral, leading to the formula:[39] {estilo de exibição 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, discutido acima, 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] Equivalentemente, 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, Contudo, depends on a choice of a Weyl chamber in {estilo de exibição {mathfrak {uma}}^{*}} , 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] {estilo de exibição 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 {estilo de exibição A(s,lambda )pi_{lambda }(g)=pi_{slambda }(g)UMA(s,lambda ).} The key property of the intertwining operators and their integrals is the multiplicative cocycle property[42] {estilo de exibição A(s_{1}s_{2},lambda )=A(s_{1},s_{2}lambda )UMA(s_{2},lambda ),} em qualquer momento {ell de estilo de exibição (s_{1}s_{2})=ell (s_{1})+bem (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 {estilo de exibição {mathfrak {uma}}_{+}^{*}} onto {estilo de exibição -{mathfrak {uma}}_{+}^{*}} . By Harish-Chandra's integral formula, it corresponds to Harish-Chandra's c-function: {estilo de exibição c(lambda )=c_{s_{0}}(lambda ).} The c-functions are in general defined by the equation {estilo de exibição 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: {estilo de exibição c_{s_{1}s_{2}}(lambda )=c_{s_{1}}(s_{2}lambda )c_{s_{2}}(lambda )} forneceu {ell de estilo de exibição (s_{1}s_{2})=ell (s_{1})+bem (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 {estilo de exibição {mathfrak {g}}_{pm alpha }} where α lies in Σ0+.[43] Then Gα is a real semisimple Lie group with real rank one, ou seja. 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: {estilo de exibição c_{s_{alfa }}(lambda )=c_{0}{fratura {2^{-eu(lambda ,alfa _{0})}Gama (eu(lambda ,alfa _{0}))}{Gama !deixei({fratura {1}{2}}deixei({fratura {1}{2}}m_{alfa }+1+eu(lambda ,alfa _{0})certo)certo)Gama !deixei({fratura {1}{2}}deixei({fratura {1}{2}}m_{alfa }+m_{2alfa }+eu(lambda ,alfa _{0})certo)certo)}},} Onde {estilo de exibição c_{0}=2^{m_{alfa }/2+m_{2alfa }}Gama !deixei({tfrac {1}{2}}(m_{alfa }+m_{2alfa }+1)certo).} 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 {estilo de exibição |{tilde {f}}(lambda )|leq C_{N}(1+|lambda |)^{-N}e^{Rleft|nome do operador {Eu estou} lambda certo|}.} 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) página. 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 {estilo de exibição {mathfrak {uma}}^{*}} ; the integral can be shifted to {estilo de exibição {mathfrak {uma}}^{*}+imu t} for μ in {estilo de exibição {mathfrak {uma}}_{+}^{*}} 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 {estilo de exibição T(f)=int_{{mathfrak {uma}}_{+}^{*}}{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 {estilo de exibição T(f)=Cf(o).} By applying this result to {estilo de exibição f_{1}(g)=int_{K}f(x^{-1}kg),dk,} segue que {displaystyle Cf=int _{{mathfrak {uma}}_{+}^{*}}{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 {estilo de exibição {mathfrak {uma}}} .[49][50] Schwartz functions The Harish-Chandra Schwartz space can be defined as[51] {estilo de exibição {matemática {S}}(Kbackslash G/K)= esquerda{fin C^{infty }(G/K)^{K}:e aí _{x}deixei|(1+d(x,o))^{m}(Delta +I)^{n}f(x)certo|

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