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.

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. 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 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 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 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. 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

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, so by the Gelfand isomorphism can be identified with the continuous functions vanishing at infinity on its spectrum X. Points in the spectrum are given by continuous *-homomorphisms of into C, i.e. characters of. If S' denotes the commutant of a set of operators S on H, then can be identified with the commutant of the regular representation of G on H. Now 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 measure μ on the locally compact space X and a unitary transformation U between H0 and L2(X, μ) which carries the operators in onto the corresponding multiplication operators. The transformation U is called the spherical Fourier transform or sometimes just the spherical transform and μ is called the Plancherel measure. The Hilbert space H0 can be identified with L2(K\G/K), the space of K-biinvariant square integrable functions on G. The characters χλ of (i.e. the points of X) can be described by positive definite spherical functions φλ on G, via the formula for f in Cc(K\G/K), where π(f) denotes the convolution operator in and the integral is with respect to Haar measure on G. The spherical functions φλ on G are given by Harish-Chandra's formula: In this formula: It follows that

Spherical principal series

The spherical function φλ can be identified with the matrix coefficient of the spherical principal series of G. If M is the centraliser 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 for b in B by where The functions f can be identified with functions in L2(K / M) and As proved, the representations of the spherical principal series are irreducible and two representations πλ and πμ 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 by Möbius transformations. The complex matrix acts as The stabiliser of the point j is the maximal compact subgroup K = SU(2), so that It carries the G-invariant Riemannian metric with associated volume element and Laplacian operator Every point in 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: The integral of an SU(2)-invariant function is given by 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 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 by the spherical transform and the spherical inversion formula Taking with fi in Cc(G / K) and, and evaluating at i yields the Plancherel formula For biinvariant functions this establishes the Plancherel theorem for spherical functions: the map is unitary and sends the convolution operator defined by into the multiplication operator defined by \tilde{f}. The spherical function Φλ is an eigenfunction of the Laplacian: Schwartz functions on R are the spherical transforms of functions f belonging to the Harish-Chandra Schwartz space By the Paley-Wiener theorem, the spherical transforms of smooth SU(2)-invariant functions of compact support are precisely functions on R which are restrictions of holomorphic functions on C satisfying an exponential growth condition As a function on G, Φλ is the matrix coefficient of the spherical principal series defined on L2(C), where C is identified with the boundary of. The representation is given by the formula The function is fixed by SU(2) and The representations πλ are irreducible and unitarily equivalent only when the sign of λ is changed. The map W of onto L2([0,∞) × C) (with measure λ2 dλ on the first factor) given by is unitary and gives the decomposition of 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 by Möbius transformations. The real matrix acts as The stabiliser of the point i is the maximal compact subgroup K = SO(2), so that = G / K. It carries the G-invariant Riemannian metric with associated area element and Laplacian operator Every point in 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: The integral of an SO(2)-invariant function is given by There are several methods for deriving the corresponding eigenfunction expansion for this ordinary differential equation including: 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 and then where Δn is the Laplacian on. 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 The adjoint operator M1* defined by satisfies The adjoint M0*, defined by averaging Mf over SO(2), satisfies for SU(2)-invariant functions F and SO(2)-invariant functions f. It follows that The function is SO(2)-invariant and satisfies On the other hand, since the integral can be computed by integrating around the rectangular indented contour with vertices at ±R and ±R + πi. Thus the eigenfunction 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. It follows that Similarly it follows that If the spherical transform of an SO(2)-invariant function on is defined by then Taking f=M1F, the SL(2, C) inversion formula for F immediately yields the spherical inversion formula for SO(2)-invariant functions on. As for SL(2,C), this immediately implies the Plancherel formula for fi in Cc(SL(2,R) / SO(2)): The spherical function φλ is an eigenfunction of the Laplacian: Schwartz functions on R are the spherical transforms of functions f belonging to the Harish-Chandra Schwartz space 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 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. As a function on G, φλ is the matrix coefficient of the spherical principal series defined on L2(R), where R is identified with the boundary of. The representation is given by the formula The function is fixed by SO(2) and The representations πλ are irreducible and unitarily equivalent only when the sign of λ is changed. The map with measure on the first factor, is given by the formula is unitary and gives the decomposition of 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. 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 1 with the group action given by Thus So on the hyperboloid, gt 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 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 where and For an SU(2)-invariant function F on H3 and an SO(2)-invariant function on H2, regarded as functions of r or t, If f(b,x) is a function on H2, Ef is defined by Thus If f is SO(2)-invariant, then, regarding f as a function of r or t, On the other hand, Thus, setting Sf(t) = f(2t), leading to the fundamental descent relation of Flensted-Jensen for M0 = ES: 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 gs. On the other hand, for F a suitable function on H3, the function QF defined by is independent of the y variable. A straightforward change of variables shows that 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 MF can be defined by The integral formula above then yields and hence, since for f SO(2)-invariant, the following adjoint formula: As a consequence Thus, as in the case of Hadamard's method of descent. with and It follows that Taking f=MF, the SL(2,C) inversion formula for F then immediately yields

Abel's integral equation

The spherical function φλ is given by so that Thus so that defining F by the spherical transform can be written The relation between F and f is classically inverted by the Abel integral equation: In fact The relation between F and \tilde{f} is inverted by the Fourier inversion formula: Hence This gives the spherical inversion for the point i. Now for fixed g in SL(2,R) define another rotation invariant function on with f1(i)=f(g(i)). On the other hand, for biinvariant functions f, so that where w = g(i). Combining this with the above inversion formula for f1 yields the general spherical inversion formula:

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 * without using Harish-Chandra's expansion of the spherical functions φλ in 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 and are their Lie algebras, then Let T be a maximal torus in U with Lie algebra Then setting there is the Cartan decomposition: The finite-dimensional irreducible representations πλ of U are indexed by certain λ in. The corresponding character formula and dimension formula of Hermann Weyl give explicit formulas for These formulas, initially defined on and, extend holomorphic to their complexifications. Moreover, where W is the Weyl group W =N_U(T)/T and δ(eX) is given by a product formula (Weyl's denominator formula) which extends holomorphically to the complexification of. 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 where. The spherical functions of G are labelled by λ in and given by the Harish-Chandra-Berezin formula 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 and the spherical inversion formula by where is a Weyl chamber. In fact the result follows from the Fourier inversion formula on since so that is just the Fourier transform of. 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 χλ/d(λ) indexed by lattice points in the interior of and the role of A is played by T. The spherical transform of f of a class function on U is given by and the spherical inversion formula now follows from the theory of Fourier series on T: There is an evident duality between these formulas and those for the non-compact dual.

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 where A+ is the image of the closure of a Weyl chamber in under the exponential map. Moreover, Since 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 and define Mf in by 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 Δc be the Laplacian on G/U. Then For F in, define MF in by Then M and M satisfy the duality relations In particular 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 is proportional to φλ on G0, the constant of proportionality being given by Moreover, in this case If f = M*F, then the spherical inversion formula for F on G implies that for f on G0: since The direct calculation of the integral for b(λ), generalising the computation of for SL(2,R), was left as an open problem by. An explicit product formula for b(λ) was known from the prior determination of the Plancherel measure by, giving where α ranges over the positive roots of the root system in 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 denote its Lie algebra. Let K be a maximal compact subgroup given as the subgroup of fixed points of a Cartan involution σ. Let be the ±1 eigenspaces of σ in, so that is the Lie algebra of K and give the Cartan decomposition Let be a maximal Abelian subalgebra of and for α in let If α ≠ 0 and, then α is called a restricted root and is called its multiplicity. Let A = exp, so that G = KAK.The restriction of the Killing form defines an inner product on and hence , which allows to be identified with. With respect to this inner product, the restricted roots Σ give a root system. Its Weyl group can be identified with. A choice of positive roots defines a Weyl chamber. The reduced root system Σ0 consists of roots α such that α/2 is not a root. Defining the spherical functions φλ as above for λ in, the spherical transform of f in Cc∞(K \ G / K) is defined by The spherical inversion formula states that where Harish-Chandra's c-function c(λ) is defined by with and the constant c0 chosen so that c(−iρ) = 1 where The Plancherel theorem for spherical functions states that the map is unitary and transforms convolution by 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 \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 \mathfrak a, invariant under W, called the radial part of the Laplacian. In general if X is in, it defines a first order differential operator (or vector field) by L can be expressed in terms of these operators by the formula where Aα in is defined by and is the Laplacian on, corresponding to any choice of orthonormal basis (Xi). Thus where 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: and therefore of L, when viewed as a W-invariant function on. 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 with Λ the cone of all non-negative integer combinations of positive roots, and the transforms of fλ under W. The expansion 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, 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. It follows that φλ can be expressed in terms as a linear combination of fλ and its transforms under W: Here c(λ) is Harish-Chandra's c-function. It describes the asymptotic behaviour of φλ in, since for X in and t > 0 large. Harish-Chandra obtained a second integral formula for φλ and hence c(λ) using the Bruhat decomposition of G: where B = MAN and the union is disjoint. Taking the Coxeter element s0 of W, the unique element mapping onto, 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 can be transferred to σ(N): for X in. Since for X in, the asymptotic behaviour of φλ can be read off from this integral, leading to the formula:

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 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). 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, 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 where σ is the Cartan involution. It satisfies the intertwining relation The key property of the intertwining operators and their integrals is the multiplicative cocycle property whenever 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 onto. By Harish-Chandra's integral formula, it corresponds to Harish-Chandra's c-function: The c-functions are in general defined by the equation 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: provided 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. In fact the integral involves only the closed connected subgroup Gα corresponding to the Lie subalgebra generated by where α lies in Σ0+. 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: This yields the following formula: where 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 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.

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 follows because the integrand defining the inverse transform extends to a meromorphic function on the complexification of ; the integral can be shifted to for μ in 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. This part of the Paley-Wiener theorem shows that 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 By applying this result to it follows that A further scaling argument allows the inequality C = 1 to be deduced from the Plancherel theorem and Paley-Wiener theorem on.

Schwartz functions

The Harish-Chandra Schwartz space can be defined as Under the spherical transform it is mapped onto the space of W-invariant Schwartz functions on 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.