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.
In the case of hyperbolic space, these expansions were known from prior results of Mehler, Weyl and Fock.
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.
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.
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.
Their work finally enabled Harish-Chandra to complete his proof of the Plancherel theorem for spherical functions in 1966.
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.
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
There can only be one such solution either because the Wronskian of the ordinary differential equation must vanish or by expanding as a power series in sinh r.[14] It follows that
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
Hadamard's method of descent relied on functions invariant under the action of 1-parameter subgroup of translations in the y parameter in
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
The direct calculation of the integral for b(λ), generalising the computation of Godement (1957) for SL(2,R), was left as an open problem by Flensted-Jensen (1978).
[30] An explicit product formula for b(λ) was known from the prior determination of the Plancherel measure by Harish-Chandra (1966), giving[31][32]
, 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
with Λ the cone of all non-negative integer combinations of positive roots, and the transforms of fλ under W. The expansion
, 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.
Although it was originally introduced by Harish-Chandra in the asymptotic expansions of spherical functions, discussed above, it was also soon understood to be intimately related to intertwining operators between induced representations, first studied in this context by Bruhat (1956).
[40] Equivalently, since ξ0 is up to scalar multiplication the unique vector fixed by K, it is an eigenvector of the intertwining operator with eigenvalue cs(λ).
, 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]
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
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.
[6][7][52] Anker (1991) found a short and simple proof, allowing the result to be deduced directly from versions of the Paley-Wiener and spherical inversion formula.