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., historical notes on the Plancherel theorem for spherical functions 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., section 21
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.The spectrum coincides with that of the commutative Banach *-algebra of integrable K-biinvariant functions on G under convolution, a dense *-subalgebra of . Points in the spectrum are given by continuous *-homomorphisms of into C, i.e. characters of .
If S denotes the 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 uniqueThe measure class of μ in the sense of the Radon–Nikodym theorem is 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:
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In this formula:
the integral is with respect to Haar measure on K; λ is an element of A* =Hom(A,T) where A is the Abelian vector subgroup in the Iwasawa decomposition G =KAN of G; λ' is defined on G by first extending λ to a character of the solvable subgroup AN, using the group homomorphism onto A, and then setting for k in K and x in AN, where 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 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
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 .
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 (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 classical spectral theory of ordinary differential equations applied to the hypergeometric equation (Mehler, Weyl, Fock); variants of Hadamard's method of descent, realising 2-dimensional hyperbolic space as the quotient of 3-dimensional hyperbolic space by the free action of a 1-parameter subgroup of SL(2,C); Abel's integral equation, following Selberg and Godement; orbital integrals (Harish-Chandra, Gelfand & Naimark).
The second and third technique will be described below, with two different methods of descent: the classical one due Hadamard, familiar from treatments of the heat equation 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 M*f 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