Plancherel theorem for spherical functions explained

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 . 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 .

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 and Gelfand & Naimark 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 introduced his celebrated c-function c(λ) to describe the asymptotic behaviour of the spherical functions φ_λ 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 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.^[2]

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 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 implicitly invokes the spherical transform; it was 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

ak{A} , so by the Gelfand isomorphism can be identified with the continuous functions vanishing at infinity on its spectrum X.^[3] Points in the spectrum are given by continuous *-homomorphisms of ak{A} into C, i.e. characters of ak{A} .

If S denotes the commutant of a set of operators S on H, then

ak{A} ^{\prime} can be identified with the commutant of the regular representation of G on H. Now ak{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^[4] measure μ on the locally compact space X and a unitary transformation U between H₀ and L²(X, μ) which carries the operators in ak{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(K\G/K), the space of K-biinvariant square integrable functions on G.

The characters χ_λ of

ak{A} (i.e. the points of X) can be described by positive definite spherical functions φ_λ on G, via the formula$$\backslash chi\_\backslash lambda(\backslash pi(f))\; =\; \backslash int\_G\; f(g)\backslash cdot\; \backslash varphi\_\backslash lambda(g)\; \backslash ,\; dg.$$for f in C_c(K\G/K), where π(f) denotes the convolution operator in ak{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:

\varphi _{λ(g)} = \int _{K} λ ^{\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 $$\backslash lambda\text{'}(kx)\; =\; \backslash Delta\_(x)^\; \backslash lambda(x)$$ 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.

Spherical principal series

See also: Principal series representation. The spherical function φ_λ 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 π_λ 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$$f(gb)=\backslash Delta(b)^\; \backslash lambda(b)\; f(g)$$for b in B by$$\backslash pi(g)f(x)=f(g^x),$$where

The functions f can be identified with functions in L²(K / M) and$$\backslash chi\_\backslash lambda(g)=(\backslash pi(g)1,1).$$

As proved, the representations of the spherical principal series are irreducible and two representations π_λ and π_μ are unitarily equivalent if and only if μ = σ(λ) for some σ in the Weyl group of A.

Example: SL(2, C)

The group G = SL(2,C) acts transitively on the quaternionic upper half space$$\backslash mathfrak^3=\backslash$$by Möbius transformations. The complex matrixacts as$$g(w)\; =\; (aw+b)(cw\; +d)^.$$

The stabiliser of the point j is the maximal compact subgroup K = SU(2), so that

ak{H} ^{3} = G/K. It carries the G-invariant Riemannian metric

$$ds^2\; =r^\backslash left(dx^2\; +\; dy^2\; +\; dr^2\backslash right)$$

with associated volume element

$$dV=\; r^\backslash ,\; dx\backslash ,dy\backslash ,dr$$

and Laplacian operator

$$\backslash Delta\; =\; -r^2\; (\backslash partial\_x^2\; +\; \backslash partial\_y^2\; +\; \backslash partial\_r^2)\; +\; r\backslash partial\_r.$$

Every point in

ak{H} ^{3} can be written as k(e^tj) 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:$$\backslash Delta=\; -\; \backslash partial\_t^2\; -\; 2\; \backslash coth\; t\; \backslash partial\_t.$$

The integral of an SU(2)-invariant function is given by$$\backslash int\; f\; \backslash ,\; dV=\; \backslash int\_^\; f(t)\; \backslash ,\; \backslash sinh^2\; t\; \backslash ,\; dt.$$

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

$$U^*\backslash Delta\; U\; =\; -\; +\; 1.$$

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

$$\backslash Phi\_\backslash lambda(t)=,$$

by the spherical transform

$$\backslash tilde(\backslash lambda)=\backslash int\; f\; \backslash Phi\_\; \backslash ,\; dV$$

and the spherical inversion formula

$$f(x)=\backslash int\; \backslash tilde(\backslash lambda)\backslash Phi\_\backslash lambda(x)\; \backslash lambda^2\; \backslash ,d\backslash lambda.$$

Taking

f = f ^{*} _{2} \star f _{1} with f_i in C_c(G / K) and f ^{*(g)} = \overline{f(g ^{-1} )} , and evaluating at i yields the Plancherel formula

$$\backslash int\_G\; f\_1\backslash overline\; \backslash ,\; dg\; =\; \backslash int\; \backslash tilde\_1(\backslash lambda)\; \backslash overline\backslash ,\; \backslash lambda^2\; \backslash ,\; d\backslash lambda.$$

For biinvariant functions this establishes the Plancherel theorem for spherical functions: the map

$$\backslash beginU:\; L^2(K\backslash backslash\; G/K)\; \backslash to\; L^2(\backslash R,\; \backslash lambda^2\backslash ,d\backslash lambda)\; \backslash \backslash U:\; f\backslash longmapsto\; \backslash tilde\backslash end$$

is unitary and sends the convolution operator defined by

f\in L ^{1(K\backslash} G/K) into the multiplication operator defined by \tilde{f} .

The spherical function Φ_λ is an eigenfunction of the Laplacian:

$$\backslash Delta\; \backslash Phi\_\backslash lambda=\; (\backslash lambda^2\; +\; 1)\; \backslash Phi\_\backslash lambda.$$

Schwartz functions on R are the spherical transforms of functions f belonging to the Harish-Chandra Schwartz space

$$\backslash mathcal\; =\; \backslash left\; \backslash \; \backslash right.\; .$$

By the Paley-Wiener theorem, the spherical transforms of smooth SU(2)-invariant functions of compact support are preciselyfunctions on R which are restrictions of holomorphic functions on C satisfying an exponential growth condition

$$|F(\backslash lambda)|\backslash le\; C\; e^.$$

As a function on G, Φ_λ is the matrix coefficient of the spherical principal series defined on L²(C), where C is identified with the boundary of

ak{H} ^{3} . The representation is given by the formula

$$\backslash pi\_\backslash lambda(g^)\backslash xi(z)=|cz\; +\; d|^\; \backslash xi(g(z)).$$

The function

$$\backslash xi\_0(z)=\backslash pi^\; \backslash left\; (1+|z|^2\; \backslash right)^$$

is fixed by SU(2) and

$$\backslash Phi\_\backslash lambda(g)=(\backslash pi\_\backslash lambda(g)\backslash xi\_0,\backslash xi\_0).$$

The representations π_λ are irreducible and unitarily equivalent only when the sign of λ is changed. The map W of

L ^{2(ak{H}} ^{3)} onto (with measure λ² dλ on the first factor) given by

$$Wf(\backslash lambda,z)=\; \backslash int\_\; f(g)\; \backslash pi\_\backslash lambda(g)\backslash xi\_0(z)\; \backslash ,\; dg$$

is unitary and gives the decomposition of

L ^{2({akH}} ^{3)} as a direct integral of the spherical principal series.

Example: SL(2, R)

The group G = SL(2,R) acts transitively on the Poincaré upper half plane

$$\backslash mathfrak^2\; =\; \backslash$$

by Möbius transformations. The real matrix

acts as

$$g(w)=(aw+b)(cw\; +d)^.$$

The stabiliser of the point i is the maximal compact subgroup K = SO(2), so that

ak{H} ^{2} = G / K.It carries the G-invariant Riemannian metric

$$ds^2\; =r^\; \backslash left(dx^2\; +\; dr^2\backslash right)$$

with associated area element

$$dA=\; r^\backslash ,\; dx\backslash ,dr$$

and Laplacian operator

$$\backslash Delta\; =\; -r^2(\backslash partial\_x^2\; +\; \backslash partial\_r^2).$$

Every point in

ak{H} ^{2} can be written as k(e^t 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:$$\backslash Delta=\; -\; \backslash partial\_t^2\; -\; \backslash coth\; t\; \backslash partial\_t.$$

The integral of an SO(2)-invariant function is given by$$\backslash int\; f\; \backslash ,\; dA=\; \backslash int\_^\; f(t)\; \backslash left|\backslash sinh\; t\backslash right|\; dt.$$

There are several methods for deriving the corresponding eigenfunction expansion for this ordinary differential equation including:

1. the classical spectral theory of ordinary differential equations applied to the hypergeometric equation (Mehler, Weyl, Fock);
2. 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);
3. Abel's integral equation, following Selberg and Godement;
4. 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 and the wave equation on hyperbolic space; and Flensted-Jensen's method on the hyperboloid.

Hadamard's method of descent

If f(x,r) is a function on

ak{H} ^{2} and

$$M\_1f(x,y,r)=r^\backslash cdot\; f(x,r)$$

then

$$\backslash Delta\_3\; M\_1\; f=\; M\_1\; \backslash left\; (\backslash Delta\_2\; +\; \backslash tfrac\; \backslash right)f,$$

where Δ_n is the Laplacian on

{ ak 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

$$\backslash Delta\_3\; M\_0=\; M\_0\; \backslash left\; (\backslash Delta\_2\; +\; \backslash tfrac\; \backslash right).$$

The adjoint operator M₁* defined by

$$M\_1^*\; F(x,r)=r^\; \backslash int\_^\backslash infty\; F(x,y,r)\backslash ,\; dy$$

satisfies

$$\backslash int\_\; (M\_1f)\backslash cdot\; F\; \backslash ,dV\; =\; \backslash int\_\; f\backslash cdot\; (M\_1^*F)\; \backslash ,dA.$$

The adjoint M₀*, defined by averaging M*f over SO(2), satisfies$$\backslash int\_\; (M\_0f)\backslash cdot\; F\; \backslash ,dV\; =\; \backslash int\_\; f\backslash cdot\; (M\_0^*F)\; \backslash ,dA$$for SU(2)-invariant functions F and SO(2)-invariant functions f. It follows that$$M\_i^*\; \backslash Delta\_3=\; \backslash left\; (\backslash Delta\_2\; +\; \backslash tfrac\; \backslash right)M\_i^*.$$

The function$$f\_\backslash lambda=\; M\_1^*\; \backslash Phi\_\backslash lambda$$is SO(2)-invariant and satisfies$$\backslash Delta\_2\; f\_\backslash lambda=\; \backslash left\; (\backslash lambda^2\; +\; \backslash tfrac\; \backslash right)f\_\backslash lambda.$$

On the other hand,$$b(\backslash lambda)=f\_\backslash lambda(i)=\backslash int\; \backslash ,\; dt=\; \backslash tanh,$$

since the integral can be computed by integrating

e ^{iλ} /\sinh t around the rectangular indented contour with vertices at ±R and ±R + πi. Thus the eigenfunction

$$\backslash phi\_\backslash lambda=b(\backslash lambda)^\; M\_1\backslash Phi\_\backslash lambda$$

satisfies the normalisation condition φ_λ(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. It follows that

$$\backslash varphi\_\backslash lambda(e^t\; i)\; =\; \backslash frac\; \backslash int\_0^\; \backslash left(\backslash cosh\; t\; -\; \backslash sinh\; t\; \backslash cos\; \backslash theta\backslash right)^\; \backslash ,\; d\backslash theta.$$

Similarly it follows that

$$\backslash Phi\_\backslash lambda\; =\; M\_1\; \backslash phi\_\backslash lambda.$$

If the spherical transform of an SO(2)-invariant function on

ak{H} ^{2} is defined by

$$\backslash tilde(\backslash lambda)=\backslash int\; f\; \backslash varphi\_\; \backslash ,dA,$$

then

$$^\backslash sim(\backslash lambda)\; =\; \backslash tilde(\backslash lambda).$$

Taking f=M₁*F, the SL(2, C) inversion formula for F immediately yields

$$f(x)=\backslash int\_^\backslash infty\; \backslash varphi\_\backslash lambda(x)\; \backslash tilde(\backslash lambda)\; \backslash tanh\; \backslash left\; (\backslash right)\; \backslash ,\; d\backslash lambda,$$

the spherical inversion formula for SO(2)-invariant functions on

ak{H} ^{2} .

As for SL(2,C), this immediately implies the Plancherel formula for f_i in C_c(SL(2,R) / SO(2)):

$$\backslash int\_\; f\_1\backslash overline\; \backslash ,dA\; =\; \backslash int\_^\backslash infty\; \backslash tilde\_1\; \backslash overline\; \backslash tanh\; \backslash left\; (\backslash right)\backslash ,\; d\backslash lambda.$$

The spherical function φ_λ is an eigenfunction of the Laplacian:

$$\backslash Delta\_2\; \backslash varphi\_\backslash lambda=\; \backslash left\; (\backslash lambda^2\; +\; \backslash tfrac\; \backslash right)\; \backslash varphi\_\backslash lambda.$$

Schwartz functions on R are the spherical transforms of functions f belonging to the Harish-Chandra Schwartz space

$$=\; \backslash left\; \backslash \; \backslash right.\; .$$

The spherical transforms of smooth SO(2)-invariant functions of compact support are precisely functions on R which are restrictions of holomorphic functions on C satisfying an exponential growth condition

$$|F(\backslash lambda)|\backslash le\; C\; e^.$$

Both these results can be deduced by descent from the corresponding results for SL(2,C), by verifying directly that the spherical transform satisfies the given growth conditions and then using the relation

(M ^{*} _{1} F) ^{\sim} = \tilde{F} .

As a function on G, φ_λ is the matrix coefficient of the spherical principal series defined on L²(R), where R is identified with the boundary of

ak{H} ^{2} . The representation is given by the formula

$$\backslash pi\_\backslash lambda(g^)\backslash xi(x)=|cx\; +\; d|^\; \backslash xi(g(x)).$$

The function

$$\backslash xi\_0(x)\; =\; \backslash pi^\; \backslash left(1+|x|^2\backslash right)^$$

is fixed by SO(2) and

$$\backslash Phi\_\backslash lambda(g)=(\backslash pi\_\backslash lambda(g)\backslash xi\_0,\backslash xi\_0).$$

The representations π_λ are irreducible and unitarily equivalent only when the sign of λ is changed. The map

W: L ^{2({akH}} ^{2)} \to L ^{2} ([ 0 , inf ty) x \R) with measure $\backslash frac\; \backslash tanh\; \backslash left(\backslash frac\; \backslash right)\backslash ,\; d\backslash lambda$ on the first factor, is given by the formula

$$Wf(\backslash lambda,x)=\; \backslash int\_\; f(g)\; \backslash pi\_\backslash lambda(g)\backslash xi\_0(x)\; \backslash ,\; dg$$

is unitary and gives the decomposition of

L ^{2(ak{H}} ^{2)} as a direct integral of the spherical principal series.

Flensted–Jensen's method of descent

Hadamard's method of descent relied on functions invariant under the action of 1-parameter subgroup of translations in the y parameter in

ak{H} ^{3} . Flensted - Jensen's method uses the centraliser of SO(2) in SL(2,C) which splits as a direct product of SO(2) and the 1-parameter subgroup K₁ of matrices

The symmetric space SL(2,C)/SU(2) can be identified with the space H³ of positive 2×2 matrices A with determinant 1with the group action given by$$g\backslash cdot\; A\; =\; gAg^*.$$

Thus

a ^{2=1+b} ^{2} + x ^{2} + y ^{2} , g_t only changes the coordinates y and a. Similarly the action of SO(2) acts by rotation on the coordinates (b,x) leaving a and y unchanged. The space H² of real-valued positive matrices A with y = 0 can be identified with the orbit of the identity matrix under SL(2,R). Taking coordinates (b,x,y) in H³ and (b,x) on H² the volume and area elements are given by

$$dV\; =(1+r^2)^\; \backslash ,db\backslash ,\; dx\backslash ,\; dy,\backslash ,\backslash ,\backslash ,\; dA=\; (1+r^2)^\; \backslash ,db\backslash ,\; dx,$$

where r² equals b² + x² + y² or b² + x²,so that r is related to hyperbolic distance from the origin by

r = \sinh t .

The Laplacian operators are given by the formula

$$\backslash Delta\_n=-L\_n\; -\; R\_n^2\; -(n-1)R\_n,\; \backslash ,$$

where

$$L\_2=\backslash partial\_b^2\; +\backslash partial\_x^2,\; \backslash ,\backslash ,\backslash ,\; R\_2=b\backslash partial\_b\; +\; x\backslash partial\_x$$

and

$$L\_3=\backslash partial\_b^2\; +\backslash partial\_x^2\; +\backslash partial\_y^2,\backslash ,\backslash ,\; \backslash ,\; R\_3=b\backslash partial\_b\; +\; x\backslash partial\_x\; +\; y\backslash partial\_y.\; \backslash ,$$

For an SU(2)-invariant function F on H³ and an SO(2)-invariant function on H², regarded as functions of r or t,

$$\backslash int\_\; F\; \backslash ,dV\; =4\backslash pi\; \backslash int\_^\backslash infty\; F(t)\; \backslash sinh^2\; t\; \backslash ,\; dt,\backslash ,\backslash ,\backslash ,\; \backslash int\_\; f\; \backslash ,dV\; =2\backslash pi\backslash int\_^\backslash infty\; f(t)\; \backslash sinh\; t\; \backslash ,\; dt.$$

If f(b,x) is a function on H², Ef is defined by

$$Ef(b,x,y)=f(b,x).\; \backslash ,$$

Thus

$$\backslash Delta\_3\; Ef\; =\; E(\backslash Delta\_2\; -\; R\_2)f.\; \backslash ,$$

If f is SO(2)-invariant, then, regarding f as a function of r or t,

$$(-\backslash Delta\_2\; +R\_2)f=\; \backslash partial\_t^2\; f\; +\; \backslash coth\; t\; \backslash partial\; \_t\; f\; +\; r\backslash partial\_r\; f\; =\backslash partial\_t^2\; f\; +\; (\backslash coth\; t\; +\; \backslash tanh\; t)\backslash partial\_t\; f.$$

On the other hand,$$\backslash partial\_t^2\; +\; (\backslash coth\; t\; +\; \backslash tanh\; t)\backslash partial\_t\; =\backslash partial\_t^2\; +\; 2\; \backslash coth(2t)\; \backslash partial\_t.$$

Thus, setting Sf(t) = f(2t),$$(\backslash Delta\_2\; -R\_2)Sf\; =\; 4S\backslash Delta\_2f,$$leading to the fundamental descent relation of Flensted-Jensen for M₀ = ES:$$\backslash Delta\_3\; M\_0\; f\; =\; 4M\_0\backslash Delta\_2\; f.$$

The same relation holds with M₀ by M, where Mf is obtained by averaging M₀f over SU(2).

The extension Ef is constant in the y variable and therefore invariant under the transformations g_s. On the other hand, for F a suitable function on H³, the function QF defined by$$QF\; =\; \backslash int\_F\backslash circ\; g\_s\; \backslash ,\; ds$$is independent of the y variable. A straightforward change of variables shows that$$\backslash int\_F\; \backslash ,\; dV\; =\; \backslash int\_\; (1+b^2\; +x^2)^\; QF\; \backslash ,\; dA.$$

Since K₁ commutes with SO(2), QF is SO(2)--invariant if F is, in particular if F is SU(2)-invariant. In this case QF is a function of r or t, so that M*F can be defined by$$M^*F(t)=QF(t/2).$$

The integral formula above then yields$$\backslash int\_\; F\; \backslash ,\; dV\; =\; \backslash int\_\; M^*F\; \backslash ,\; dA$$and hence, since for f SO(2)-invariant,$$M^*((Mf)\backslash cdot\; F)\; =f\; \backslash cdot\; (M^*F),$$the following adjoint formula:$$\backslash int\_\; (Mf)\backslash cdot\; F\; \backslash ,\; dV\; =\backslash int\_\; f\backslash cdot\; (M*F)\backslash ,\; dV.$$

As a consequence$$M^*\backslash Delta\_3\; =\; 4\backslash Delta\_2M^*.$$

Thus, as in the case of Hadamard's method of descent.

$$M^*\backslash Phi\_\; =b(\backslash lambda)\; \backslash varphi\_\backslash lambda$$with$$b(\backslash lambda)=M^*\backslash Phi\_(0)=\backslash pi\; \backslash tanh\; \backslash pi\; \backslash lambda$$and$$\backslash Phi\_=M\backslash varphi\_\backslash lambda.$$

It follows that$$^\backslash sim(\backslash lambda)\; =\; \backslash tilde(2\backslash lambda).$$

Taking f=M*F, the SL(2,C) inversion formula for F then immediately yields$$f(x)=\backslash int\_^\backslash infty\; \backslash varphi\_\backslash lambda(x)\; \backslash tilde(\backslash lambda)\backslash ,\; \backslash tanh\backslash left(\backslash frac\backslash right)\backslash ,\; d\backslash lambda,$$

Abel's integral equation

The spherical function φ_λ is given by$$\backslash varphi\_\backslash lambda(g)=\backslash int\_K\; \backslash alpha\text{'}(kg)\; \backslash ,\; dk,$$so that$$\backslash tilde(\backslash lambda)=\backslash int\_S\; f(s)\; \backslash alpha\text{'}(s)\backslash ,\; ds,$$

Thus$$\backslash tilde(\backslash lambda)=\backslash int\_^\backslash infty\; \backslash int\_0^\backslash infty\; f\backslash !\backslash left(\backslash frac\backslash right)a^\; \backslash ,\; da\backslash ,db,$$

so that defining F by

$$F(u)=\backslash int\_^\backslash infty\; f\backslash !\backslash left(u\; +\; \backslash frac\backslash right)\; \backslash ,\; dt,$$

the spherical transform can be written

$$\backslash tilde(\backslash lambda)=\backslash int\_0^\backslash infty\; F\backslash !\backslash left(\backslash frac\backslash right)\; a^\backslash ,\; da=\; \backslash int\_0^\backslash infty\; F(\backslash cosh\; t)e^\; \backslash ,\; dt.$$

The relation between F and f is classically inverted by the Abel integral equation:

$$f(x)=\; \backslash frac\; \backslash int\_^\backslash infty\; F\text{'}\backslash !\backslash left(x\; +\backslash right)\backslash ,\; dt.$$

In fact

$$\backslash int\_^\backslash infty\; F\text{'}\backslash !\backslash left(x\; +\; \backslash frac\backslash right)\backslash ,\; dt=\; \backslash int\_^\backslash infty\; \backslash int\_^\backslash infty\; f\text{'}\backslash !\backslash left(x\; +\; \backslash frac\backslash right)\; \backslash ,\; dt\backslash ,du=\; \backslash int\_0^\backslash infty\; f\text{'}\backslash !\backslash left(x\; +\; \backslash frac\backslash right)\; r\backslash ,\; dr=\; 2\backslash pi\; f(x).$$

The relation between F and

\tilde{f} is inverted by the Fourier inversion formula:

$$F(\backslash cosh\; t)=\; \backslash int\_0^\backslash infty\backslash tilde(i\backslash lambda)\backslash cos(\backslash lambda\; t)\; \backslash ,\; d\backslash lambda.$$

Hence

$$f(i)=\; \backslash int\_0^\backslash infty\; \backslash tilde(\backslash lambda)\; \backslash lambda\backslash ,\; d\backslash lambda\; \backslash int\_^\backslash infty\; \backslash cosh\; \backslash ,\; dt\; =\; \backslash int\_^\backslash infty\; \backslash tilde(\backslash lambda)\; \backslash tanh\backslash left(\backslash right)\backslash ,\; d\backslash lambda.$$

This gives the spherical inversion for the point i. Now for fixed g in SL(2,R) define

$$f\_1(w)=\backslash int\_K\; f(gkw)\; \backslash ,\; dk,$$

another rotation invariant function on

ak{H} ^{2} with f₁(i)=f(g(i)). On the other hand, for biinvariant functions f,

$$\backslash pi\_\backslash lambda(f)\backslash xi\_0\; =\backslash tilde(\backslash lambda)\; \backslash xi\_0$$

so that

$$\backslash tilde\_1(\backslash lambda)=\backslash tilde(\backslash lambda)\backslash cdot\; \backslash varphi\_\backslash lambda(w),$$

where w = g(i). Combining this with the above inversion formula for f₁ yields the general spherical inversion formula:

$$f(w)\; =\; \backslash int\_0^\backslash infty\; \backslash tilde(\backslash lambda)\; \backslash varphi\_\backslash lambda(w)\; \backslash tanh\backslash left(\backslash right)\backslash ,\; d\backslash lambda.$$

Other special cases

All complex semisimple Lie groups or the Lorentz groups SO⁰(N,1) with N odd can be treated directly by reduction to the usual Fourier transform. The remaining real Lorentz groups can be deduced by Flensted-Jensen's method of descent, as can other semisimple Lie groups of real rank one. Flensted-Jensen's method of descent also applies to the treatment of real semisimple Lie groups for which the Lie algebras are normal real forms of complex semisimple Lie algebras. The special case of SL(N,R) is treated in detail in ; this group is also the normal real form of SL(N,C).

The approach of applies to a wide class of real semisimple Lie groups of arbitrary real rank and yields the explicit product form of the Plancherel measure on

ak{a} * without using Harish-Chandra's expansion of the spherical functions φ_λin terms of his c-function, discussed below. Although less general, it gives a simpler approach to the Plancherel theorem for this class of groups.

Complex semisimple Lie groups

If G is a complex semisimple Lie group, it is the complexification of its maximal compact subgroup U, a compact semisimple Lie group. If

ak{g} and ak{u} are their Lie algebras, then ak{g} = ak{u} ⊕ i ak{u}. Let T be a maximal torus in U with Lie algebra ak{t}. Then setting

$$A=\; \backslash exp\; i\; \backslash mathfrak,\; \backslash qquad\; P=\; \backslash exp\; i\; \backslash mathfrak,$$

there is the Cartan decomposition:

$$G=P\backslash cdot\; U\; =\; UAU.$$

The finite-dimensional irreducible representations π_λ of U are indexed by certain λ in

ak{t} ^{*} .^[5] The corresponding character formula and dimension formula of Hermann Weyl give explicit formulas for

$$\backslash chi\_\backslash lambda(e^X)\; =\; \backslash operatorname\; \backslash pi\_\backslash lambda(e^X),\; (X\backslash in\; \backslash mathfrak),\; \backslash qquad\; d(\backslash lambda)=\backslash dim\; \backslash pi\_\backslash lambda.$$

These formulas, initially defined on

ak{t} ^{* x} ak{t} and ak{t} ^{*} , extend holomorphic to their complexifications. Moreover,

$$\backslash chi\_\backslash lambda(e^X)=,$$

W = N _{U(T)/T} and δ(e^X) is given by a product formula (Weyl's denominator formula) which extends holomorphically to the complexification of ak{t} . There is a similar product formula for d(λ), a polynomial in λ.

On the complex group G, the integral of a U-biinvariant function F can be evaluated as

$$\backslash int\_G\; F(g)\; \backslash ,\; dg\; =\; \backslash int\_\; F(e^X)\backslash ,\; |\backslash delta(e^X)|^2\; \backslash ,\; dX.$$

where

ak{a}=i ak{t} .

The spherical functions of G are labelled by λ in

ak{a}=i ak{t} ^{*} and given by the Harish-Chandra-Berezin formula

$$\backslash Phi\_\backslash lambda(e^X)\; =\; .$$

They are the matrix coefficients of the irreducible spherical principal series of G induced from the character of the Borel subgroup of G corresponding to λ; these representations are irreducible and can all be realized on L²(U/T).

The spherical transform of a U-biinvariant function F is given by

$$\backslash tilde(\backslash lambda)=\backslash int\_G\; F(g)\; \backslash Phi\_(g)\backslash ,\; dg$$

and the spherical inversion formula by

$$F(g)\; =\backslash int\_\; \backslash tilde(\backslash lambda)\backslash Phi\_\backslash lambda(g)\; |d(\backslash lambda)|^2\; \backslash ,\; d\backslash ,\backslash lambda=\backslash int\_\; \backslash tilde(\backslash lambda)\backslash Phi\_\backslash lambda(g)\; |d(\backslash lambda)|^2\; \backslash ,\; d\backslash ,\backslash lambda,$$

where

{ ak a} ^{*} _{+} is a Weyl chamber. In fact the result follows from the Fourier inversion formula on ak{a} since$$d(\backslash lambda)\backslash delta(e^X)\backslash Phi\_\backslash lambda(e^X)=\backslash sum\_\; (\backslash sigma)\; e^,$$so that \overline{d( λ )}\tilde{F}( λ ) is just the Fourier transform of F(e ^{X)\delta(e} ^{X)} .

Note that the symmetric space G/U has as compact dual the compact symmetric space U x U / U, where U is the diagonal subgroup. The spherical functions for the latter space, which can be identified with U itself, are the normalized characters χ_λ/d(λ) indexed by lattice points in the interior of

ak {a} ^{*} _{+} and the role of A is played by T. The spherical transform of f of a class function on U is given by

$$\backslash tilde(\backslash lambda)=\backslash int\_U\; f(u)\; \backslash ,du$$

and the spherical inversion formula now follows from the theory of Fourier series on T:

$$f(u)=\backslash sum\_\; \backslash tilde(\backslash lambda)\; d(\backslash lambda)^2.$$

There is an evident duality between these formulas and those for the non-compact dual.^[6]

Real semisimple Lie groups

Let G₀ be a normal real form of the complex semisimple Lie group G, the fixed points of an involution σ, conjugate linear on the Lie algebra of G. Let τ be a Cartan involution of G₀ extended to an involution of G, complex linear on its Lie algebra, chosen to commute with σ. The fixed point subgroup of τσ is a compact real form U of G, intersecting G₀ in a maximal compact subgroup K₀. The fixed point subgroup of τ is K, the complexification of K₀. Let G₀= K₀·P₀ be the corresponding Cartan decomposition of G₀ and let A be a maximal Abelian subgroup of P₀. proved that$$G\; =\; K\; A\_+\; U,$$where A₊ is the image of the closure of a Weyl chamber in

ak{a} under the exponential map. Moreover,$$K\backslash backslash\; G/U\; =\; A\_+.$$

Since$$K\_0\backslash backslash\; G\_0/K\_0\; =\; A\_+$$

it follows that there is a canonical identification between K \ G / U, K₀ \ G₀ /K₀ and A₊. Thus K₀-biinvariant functions on G₀ can be identified with functions on A₊ as can functions on G that are left invariant under K and right invariant under U. Let f be a function in

C ^{inf ty} _{c(K} _{0\backslash} G _{0} /K _{0)} and define Mf in C ^{inf ty} _{c(U\backslash} G /U) by$$Mf(a)=\backslash int\_U\; f(ua^2)\; \backslash ,du.$$

Here a third Cartan decomposition of G = UAU has been used to identify U \ G / U with A₊.

Let Δ be the Laplacian on G₀/K₀ and let Δ_c be the Laplacian on G/U. Then$$4M\backslash Delta\; =\; \backslash Delta\_c\; M.$$

For F in

C ^{inf ty} _{c(U\backslash} G /U) , define M*F in C ^{inf ty} _{c(K} _{0\backslash} G _{0} /K _{0)} by$$M^*F(a^2)=\backslash int\_K\; F(ga)\; \backslash ,\; dg.$$

Then M and M* satisfy the duality relations$$\backslash int\_\; (Mf)\; \backslash cdot\; F\; =\; \backslash int\_\; f\backslash cdot\; (M^*F).$$

In particular$$M^*\backslash Delta\_c=\; 4\backslash Delta\; M^*.$$

There is a similar compatibility for other operators in the center of the universal enveloping algebra of G₀. It follows from the eigenfunction characterisation of spherical functions that

M ^{*\Phi} _{2λ} is proportional to φ_λ on G₀, the constant of proportionality being given by$$b(\backslash lambda)=M^*\backslash Phi\_(1)=\backslash int\_K\; \backslash Phi\_(k)\backslash ,\; dk.$$

Moreover, in this case$$(M^*\; F)^\backslash sim(\backslash lambda)=\; \backslash tilde(2\backslash lambda).$$

If f = M*F, then the spherical inversion formula for F on G implies that for f on G₀:$$f(g)=\; \backslash int\_\; \backslash tilde(\backslash lambda)\; \backslash varphi\_\backslash lambda(g)\; \backslash ,\backslash ,\; 2^\backslash cdot\; |b(\backslash lambda)|\backslash cdot\; |d(2\backslash lambda)|^2\; \backslash ,d\backslash lambda,$$since$$f(g)=\; M^*F(g)=\; \backslash int\_\; \backslash tilde(2\backslash lambda)\; M^*\backslash Phi\_(g)\; 2^\; |d(2\backslash lambda)|^2\; \backslash ,\; d\backslash lambda=\; \backslash int\_\; \backslash tilde(\backslash lambda)\; \backslash varphi\_\backslash lambda(g)\; \backslash ,\backslash ,b(\backslash lambda)\; 2^\; |d(2\backslash lambda)|^2\; \backslash ,\; d\backslash lambda.$$

The direct calculation of the integral for b(λ), generalising the computation of for SL(2,R), was left as an open problem by .^[7] An explicit product formula for b(λ) was known from the prior determination of the Plancherel measure by, giving$$b(\backslash lambda)\; =C\backslash cdot\; d(2\backslash lambda)^\backslash cdot\; \backslash prod\_\; \backslash tanh,$$

where α ranges over the positive roots of the root system in

ak{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

ak{g} denote its Lie algebra. Let K be a maximal compact subgroupgiven as the subgroup of fixed points of a Cartan involution σ. Let ak{g} _{\pm} be the ±1 eigenspaces of σ in ak{g} , so that ak{k}= ak{g} _{+} is the Lie algebra of K and ak{p}= ak{g} _{-} give the Cartan decomposition

$$\backslash mathfrak=\backslash mathfrak+\backslash mathfrak,\backslash ,\backslash ,\; G=\backslash exp\; \backslash mathfrak\backslash cdot\; K.$$

Let

ak {a} be a maximal Abelian subalgebra of ak{p} and for α in ak {a} ^{*} let

$$\backslash mathfrak\_\backslash alpha=\backslash .$$

If α ≠ 0 and

ak{g} _{\alpha\ne} ( 0 ) , then α is called a restricted root and m _{\alpha} = \dim ak{g} _{\alpha} is called its multiplicity. Let A = exp ak {a} , so that G = KAK.The restriction of the Killing form defines an inner product on ak{p} and hence ak{a} , which allows ak{a} ^{*} to be identified with ak{a} . With respect to this inner product, the restricted roots Σ give a root system. Its Weyl group can be identified with W = N _{K(A)} / C _{K(A)} . A choice of positive roots defines a Weyl chamber ak{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

ak {a} ^{*} , the spherical transform of f in C_c^∞(K \ G / K) is defined by$$\backslash tilde(\backslash lambda)=\backslash int\_G\; f(g)\; \backslash varphi\_(g)\backslash ,\; dg.$$

The spherical inversion formula states that$$f(g)=\backslash int\_\; \backslash tilde(\backslash lambda)\; \backslash varphi\_\backslash lambda(g)\backslash ,\; |c(\backslash lambda)|^\backslash ,\; d\backslash lambda,$$where Harish-Chandra's c-function c(λ) is defined by$$c(\backslash lambda)=c\_0\backslash cdot\backslash prod\_\; \backslash frac$$with

\alpha ^{-1} _{0= (\alpha,\alpha)} \alpha and the constant c₀ chosen so that c(−iρ) = 1 where$$\backslash rho\; =\; \backslash frac\; \backslash sum\_\; m\_\backslash alpha\; \backslash alpha.$$

The Plancherel theorem for spherical functions states that the map$$W:f\backslash mapsto\; \backslash tilde,\backslash ,\backslash ,\backslash ,\backslash \; L^2(K\backslash backslash\; G\; /K)\; \backslash rightarrow\; L^2(\backslash mathfrak\_+^*,\; |c(\backslash lambda)|^\backslash ,\; d\backslash lambda)$$is unitary and transforms convolution by

f\in L ^{1(K\backslash} G/K) into multiplication by \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

ak 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 ak a , invariant under W, called the radial part of the Laplacian. In general if X is in ak{a} , it defines a first order differential operator (or vector field) by$$Xf(y)=\backslash left.\backslash frac\; f(y\; +\; tX)\backslash right|\_.$$

L can be expressed in terms of these operators by the formula$$L\; =\; \backslash Delta\_\; -\; \backslash sum\_\; m\_\backslash alpha\; \backslash ,\; \backslash coth\; \backslash alpha\; \backslash ,\; A\_\backslash alpha,$$where A_α in

ak{a} is defined by$$(A\_\backslash alpha,X)=\backslash alpha(X)$$and$$\backslash Delta\_\; =\; -\backslash sum\; X\_i^2$$is the Laplacian on ak{a} , corresponding to any choice of orthonormal basis (X_i).

Thus$$L=L\_0\; -\backslash sum\_\; m\_\backslash alpha\backslash ,\; (\backslash coth\; \backslash alpha\; -1)\; A\_\backslash alpha,$$where$$L\_0=\backslash Delta\_\; -\; \backslash sum\_\; A\_\backslash alpha,$$so that L can be regarded as a perturbation of the constant-coefficient operator L₀.

Now the spherical function φ_λ is an eigenfunction of the Laplacian:$$\backslash Delta\backslash varphi\_\backslash lambda=\backslash left(\backslash left\backslash |\backslash lambda\backslash right\backslash |^2\; +\; \backslash left\backslash |\backslash rho\backslash right\backslash |^2\backslash right)\backslash varphi\_\backslash lambda$$and therefore of L, when viewed as a W-invariant function on

ak{a} .

Since e^iλ–ρ and its transforms under W are eigenfunctions of L₀ with the same eigenvalue, it is natural look for a formula for φ_λ in terms of a perturbation series$$f\_\backslash lambda=e^\backslash sum\_\; a\_\backslash mu(\backslash lambda)\; e^,$$with Λ the cone of all non-negative integer combinations of positive roots, and the transforms of f_λ under W. The expansion$$\backslash coth\; x-1\; =2\; \backslash sum\_\; e^,$$

leads to a recursive formula for the coefficients a_μ(λ). In particular they are uniquely determined and theseries and its derivatives converges absolutely on

ak{a} _{+} , a fundamental domain for W. Remarkably it turns out that f_λ is also an eigenfunction of the other G-invariant differential operators on G/K, each of which induces a W-invariant differential operator on ak{a} .

It follows that φ_λ can be expressed in terms as a linear combination of f_λ and its transforms under W:$$\backslash varphi\_\backslash lambda=\backslash sum\_\; c(s\backslash lambda)\; f\_.$$

Here c(λ) is Harish-Chandra's c-function. It describes the asymptotic behaviour of φ_λ in

ak{a} _{+} , since$$\backslash varphi\_\backslash lambda(e^tX)\; \backslash sim\; c(\backslash lambda)\; e^$$for X in ak{a} _{+} and t > 0 large.

Harish-Chandra obtained a second integral formula for φ_λ and hence c(λ) using the Bruhat decomposition of G:$$G\; =\; \backslash bigcup\_\; B\; s\; B,$$

where B = MAN and the union is disjoint. Taking the Coxeter element s₀ of W, the unique element mapping

ak{a} _{+} onto - ak{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 formulafor φ_λ initially defined over K/M

$$\backslash varphi\_\backslash lambda(g)\; =\; \backslash int\_\; \backslash lambda\text{'}(gk)^\backslash ,\; dk.$$

can be transferred to σ(N):$$\backslash varphi\_\backslash lambda(e^X)\; =\; e^\; \backslash int\_\; \backslash ,\; dn,$$for X in

ak{a} .

Since$$\backslash lim\_\; e^\; n\; e^\; =\; 1$$

for X in

ak{a} _{+} , the asymptotic behaviour of φ_λ can be read off from this integral, leading to the formula:$$c(\backslash lambda)=\backslash int\_\; \backslash overline\backslash ,\; dn.$$

Harish-Chandra's c-function

See main article: article and Harish-Chandra's c-function.

The many roles of Harish-Chandra's c-function in non-commutative harmonic analysis are surveyed in . 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 . These operators exhibit the unitary equivalence between π_λ and π_sλ for s in the Weyl group and a c-function c_s(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).^[8] Equivalently, since ξ₀ is up to scalar multiplication the unique vector fixed by K, it is an eigenvector of the intertwining operator with eigenvalue c_s(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

ak{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$$A(s,\backslash lambda)F(k)=\backslash int\_\; F(ksn)\backslash ,\; dn,$$where σ is the Cartan involution. It satisfies the intertwining relation$$A(s,\backslash lambda)\backslash pi\_\backslash lambda(g)\; =\backslash pi\_(g)\; A(s,\backslash lambda).$$

The key property of the intertwining operators and their integrals is the multiplicative cocycle property$$A(s\_1s\_2,\backslash lambda)\; =\; A(s\_1,s\_2\backslash lambda)\; A(s\_2,\backslash lambda),$$whenever$$\backslash ell(s\_1s\_2)\; =\; \backslash ell(s\_1)\; +\; \backslash ell(s\_2)$$

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

ak{a} ^{*} _{+} onto - ak{a} ^{*} _{+} . By Harish-Chandra's integral formula, it corresponds to Harish-Chandra's c-function:

$$c(\backslash lambda)=c\_(\backslash lambda).$$

The c-functions are in general defined by the equation$$A(s,\backslash lambda)\backslash xi\_0\; =c\_s(\backslash lambda)\backslash xi\_0,$$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:$$c\_(\backslash lambda)\; =c\_(s\_2\; \backslash lambda)c\_(\backslash lambda)$$provided$$\backslash ell(s\_1\; s\_2)\; =\; \backslash ell(s\_1)\; +\; \backslash ell(s\_2).$$

This reduces the computation of c_s to the case when s = s_α, the reflection in a (simple) root α, the so-called "rank-one reduction" of . In fact the integral involves only the closed connected subgroup G^α corresponding to the Lie subalgebra generated by

ak{g} _{\pm} where α lies in Σ₀⁺. Then G^α is a real semisimple Lie group with real rank one, i.e. dim A^α = 1, and c_s is just the Harish-Chandra c-function of G^α. In this case the c-function can be computed directly by various means:

• by noting that φ_λ 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;
• by directly computing the integral, which can be expressed as an integral in two variables and hence a product of two beta functions.

This yields the following formula:$$c\_(\backslash lambda)\; =\; c\_0\; \backslash frac,$$where$$c\_0\; =\; 2^\; \backslash Gamma\backslash !\backslash left(\backslash tfrac\; (m\_\backslash alpha+m\_\; +1)\backslash right).$$

The general Gindikin–Karpelevich formula for c(λ) is an immediate consequence of this formula and the multiplicative properties of c_s(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

$$|\backslash tilde(\backslash lambda)|\backslash le\; C\_N\; (1+|\backslash lambda|)^\; e^.$$

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 (pg. 37).

The theorem was later proved by independently of the spherical inversion theorem, using a modification of his method of reduction to the complex case.^[9]

Rosenberg's proof of inversion formula

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 followsbecause the integrand defining the inverse transform extends to a meromorphic function on the complexification of

ak{a} ^{*} ; the integral can be shifted to ak{a} ^{*} + i\mu t for μ in ak{a} ^{*} _{+} and t > 0. Using Harish-Chandra's expansionof φ_λ 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.

This part of the Paley-Wiener theorem shows that$$T(f)=\backslash int\_\; \backslash tilde(\backslash lambda)\; |c(\backslash lambda)|^\; \backslash ,\; d\backslash lambda$$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$$T(f)\; =\; Cf(o).$$

By applying this result to$$f\_1(g)=\backslash int\_K\; f(x^kg)\backslash ,\; dk,$$it follows that$$Cf=\backslash int\_\; \backslash tilde(\backslash lambda)\; \backslash varphi\_\backslash lambda\; |c(\backslash lambda)|^\; \backslash ,\; d\backslash lambda.$$

A further scaling argument allows the inequality C = 1 to be deduced from the Plancherel theorem and Paley-Wiener theorem on

ak{a} .

Schwartz functions

The Harish-Chandra Schwartz space can be defined as

$$\backslash mathcal(K\backslash backslash\; G/K)=\; \backslash left\; \backslash .$$

Under the spherical transform it is mapped onto

l{S}( ak{a} ^{*)} ^{W,} the space of W-invariantSchwartz functions on ak{a} ^{*.}

The original proof of Harish-Chandra was a long argument by induction. found a short and simple proof, allowing the result to be deduced directly from versions of the Paley-Wiener and spherical inversion formula. He proved that the spherical transform of a Harish-Chandra Schwartz function is a classical Schwartz function. His key observation was then to show that the inverse transform was continuous on the Paley-Wiener space endowed with classical Schwartz space seminorms, using classical estimates.

References

• • • • • • • • • • • • , Appendix to Chapter VI, The Plancherel Formula for Complex Semisimple Lie Groups.
        • , section 21.
  • (a general introduction for physicists)
  • .

Notes and References

1. , historical notes on the Plancherel theorem for spherical functions
2. , section 21
3. The spectrum coincides with that of the commutative Banach *-algebra of integrable K-biinvariant functions on G under convolution, a dense *-subalgebra of ak{A} .
4. The measure class of μ in the sense of the Radon–Nikodym theorem is unique.
5. These are indexed by highest weights shifted by half the sum of the positive roots.
6. The spherical inversion formula for U is equivalent to the statement that the functions \chi _{λ} d( λ ) ^{-1/2} form an orthonormal basis for the class functions.
7. b(λ) can be written as integral over A₀ where K = K₀ A₀ K₀ is the Cartan decomposition of K. The integral then becomes an alternating sum of multidimensional Godement-type integrals, whose combinatorics is governed by that of the Cartan-Helgason theorem for U/K₀. An equivalent computation that arises in the theory of the Radon transform has been discussed by, and .
8. , Chapter VII
9. The second statement on supports follows from Flensted-Jensen's proof by using the explicit methods associated withKostant polynomials instead of the results of Mustapha Rais.