# Plancherel theorem for spherical functions

In mathematics, the **Plancherel theorem for spherical functions** is an important result in the representation theory of semisimple Lie groups, due in its final form to Harish-Chandra. It is a natural generalisation in non-commutative harmonic analysis of the Plancherel formula and Fourier inversion formula in the representation theory of the group of real numbers in classical harmonic analysis and has a similarly close interconnection with the theory of differential equations.
It is the special case for zonal spherical functions of the general Plancherel theorem for semisimple Lie groups, also proved by Harish-Chandra. The Plancherel theorem gives the eigenfunction expansion of radial functions for the Laplacian operator on the associated symmetric space *X*; it also gives the direct integral decomposition into irreducible representations of the regular representation on L^{2}(*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 Template:Harvtxt.

## History

The first versions of an abstract Plancherel formula for the Fourier transform on a unimodular locally compact group *G* were due to Segal and Mautner.^{[1]} At around the same time, Harish-Chandra^{[2]}^{[3]} and Gelfand & Naimark^{[4]}^{[5]} derived an explicit formula for SL(2,R) and complex semisimple Lie groups, so in particular the Lorentz groups. A simpler abstract formula was derived by Mautner for a "topological" symmetric space *G*/*K* corresponding to a maximal compact subgroup *K*. Godement gave a more concrete and satisfactory form for positive definite spherical functions, a class of special functions on *G*/*K*. Since when *G* is a semisimple Lie group these spherical functions 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^{[6]}^{[7]} introduced his celebrated **c-function** *c*(λ) to describe the asymptotic behaviour of the spherical functions φ_{λ} and proposed *c*(λ)^{−2} *d*λ 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 Karpelevič to derive a product formula^{[8]} 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.^{[9]}

In many special cases, for example for complex semisimple group or the Lorentz groups, there are simple methods to develop the theory directly. Certain subgroups of these groups can be treated by techniques generalising the well-known "method of descent" due to Jacques Hadamard. In particular Template:Harvtxt 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 Template:Harvtxt implicitly invokes the spherical transform; it was Template:Harvtxt 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

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Let *G* be a semisimple Lie group and *K* a maximal compact subgroup of *G*. The Hecke algebra
*C*_{c}(*K* \*G*/*K*), consisting of compactly supported *K*-biinvariant continuous functions on *G*, acts by convolution on the Hilbert space *H*=*L*^{2}(*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*.^{[10]} 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 *H*_{0} of *K*-invariant vectors in *H*. Moreover the abelian von Neumann algebra it generates on *H*_{0} is maximal Abelian. By spectral theory, there is an essentially unique^{[11]} measure μ on the locally compact space *X* and a unitary transformation *U* between *H*_{0} and *L*^{2}(*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 *H*_{0} can be identified with *L*^{2}(*K*\*G*/*K*), the space of *K*-biinvariant square integrable functions on *G*.

The characters χ_{λ} of (i.e. the points of *X*) can be described by positive definite spherical functions φ_{λ} on *G*, via the formula

for *f* in *C*_{c}(*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 φ_{λ} on *G* are given by Harish-Chandra's formula:

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 Δ_{AN}is the modular function of*AN*.

- Two different characters λ
_{1}and λ_{2}give the same spherical function if and only if λ_{1}= λ_{2}·*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

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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

for *b* in *B* by

where

The functions *f* can be identified with functions in L^{2}(*K* / *M*) and

As Template:Harvtxt 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

by Möbius transformations. The complex matrix

acts as

The stabiliser of the point **j** is the maximal compact subgroup *K* = SU(2), so that = *G* / *K*.
It carries the *G*-invariant Riemannian metric

with associated volume element

Every point in can be written as *k*(*e*^{t}**j**) 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 L^{2}(**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 *f*_{i} in C_{c}(*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 L^{1}(*K* \ *G* / *K*) into the multiplication operator defined by .

The spherical function Φ_{λ} 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*, Φ_{λ} is the matrix coefficient of the spherical principal series defined on L^{2}(**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 π_{λ} are irreducible and unitarily equivalent only when the sign of λ is changed.
The map *W* of onto L^{2}([0,∞) x**C**) (with measure λ^{2} *d*λ 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 complex 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

Every point in 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*:

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^{[12]} and the wave equation^{[13]} on hyperbolic space; and Flensted-Jensen's method on the hyperboloid.

### Hadamard's method of descent

If *f*(*x*,*r*) is a function on and

then

where Δ_{n} is the Laplacian on .

Since the action of SL(2,**C**) commutes with Δ_{3}, the operator
*M*_{0} on S0(2)-invariant functions obtained by averaging *M*_{1}*f* by the action of SU(2)
also satisfies

The adjoint operator *M*_{1}* defined by

satisfies

The adjoint *M*_{0}*, 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

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*.^{[14]}
It follows that

Similarly it follows that

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

then

Taking *f*=*M*_{1}**F*, 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 *f*_{i} in C_{c}(SL(2,**R**) / SO(2)):

The spherical function φ_{λ} 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**),^{[15]}
by verifying directly that the spherical transform satisfies the given growth conditions^{[16]}^{[17]} and then using the relation .

As a function on *G*, φ_{λ} is the matrix coefficient of the spherical principal series defined on L^{2}(**R**), where **R** is identified with the boundary of . The representation is given by the formula

The function

is fixed by S0(2) and

The representations π_{λ} are irreducible and unitarily equivalent only when the sign of λ is changed.
The map *W* of onto L^{2}([0,∞) x**R**), 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 *K*_{1} of matrices

The symmetric space SL(2,**C**)/SU(2) can be identified with the space **H**^{3} of positive 2×2 matrices *A* with determinant 1

with the group action given by

Thus

So on the hyperboloid , *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 **H**^{2} 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 **H**^{3} and (*b*,*x*) on **H**^{2} the volume and area elements are given by

where *r*^{2} equals *b*^{2} + *x*^{2} + *y*^{2} or *b*^{2} + *x*^{2},
so that *r* is related to hyperbolic distance from the origin by .

The Laplacian operators are given by the formula

where

and

For an SU(2)-invariant function *F* on **H**^{3} and an SO(2)-invariant function on **H**^{2}, regarded as
functions of *r* or *t*,

If *f*(*b*,*x*) is a function on **H**^{2}, *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*(2*t*),

leading to the fundamental *descent relation* of Flensted-Jensen for *M*_{0} = *ES*:

The same relation holds with *M*_{0} by *M*, where *Mf* is obtained by averaging *M*_{0}*f* 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 **H**^{3}, the function *QF* defined by

is independent of the *y* variable. A straightforward change of variables shows that

Since *K*_{1} 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

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*=*M***F*, the SL(2,**C**) inversion formula for *F* then immediately yields

### Abel's integral equation

The spherical function φ_{λ} 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^{[18]}

The relation between *F* and 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^{[19]}

another rotation invariant function on with *f*_{1}(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 *f*_{1} yields the general spherical inversion formula:

## Other special cases

All complex semisimple Lie groups or the Lorentz groups SO^{0}(*N*,1) with *N* odd can be treated directly by reduction to the usual Fourier transform.^{[15]}^{[20]} 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.^{[21]} 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.^{[15]} The special case of SL(N,**R**) is treated in detail in Template:Harvtxt; this group is also the normal real form of SL(N,**C**).

The approach of Template:Harvtxt 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 φ_{λ}
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 π_{λ} of *U* are indexed by certain λ in .^{[22]} 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 and δ(*e*^{X}) 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

The spherical functions of *G* are labelled by λ in and given by the Harish-Chandra-Berezin formula^{[23]}

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 L^{2}(*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^{[24]}

so that is just the Fourier transform of .

Note that the symmetric space *G* / *U* has as *compact dual*^{[25]} 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 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.^{[26]}

### Real semisimple Lie groups

Let *G*_{0} 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 *G*_{0} 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 *G*_{0} in a maximal compact subgroup *K*_{0}. The fixed point subgroup of τ is *K*, the complexification of *K*_{0}. Let *G*_{0}= *K*_{0}·*P*_{0} be the corresponding Cartan decomposition of *G*_{0} and let *A* be a maximal Abelian subgroup of *P*_{0}. Template:Harvtxt 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*, *K*_{0} \ *G*_{0} /*K*_{0} and *A*_{+}. Thus *K*_{0}-biinvariant functions on *G*_{0} 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 *G*_{0}/*K*_{0} and let Δ_{c} be the Laplacian on *G*/*U*. Then

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 *G*_{0}. It follows from
the eigenfunction characterisation of spherical functions that is proportional to φ_{λ} on *G*_{0}, the constant of proportionality being given by

Moreover in this case^{[27]}

If *f* = *M***F*, then the spherical inversion formula for *F* on *G* implies that for *f* on *G*_{0}:^{[28]}^{[29]}

since

The direct calculation of the integral for *b*(λ), generalising the computation of Template:Harvtxt for SL(2,**R**), was left as an open problem by Template:Harvtxt.^{[30]} An explicit product formula for *b*(λ) was known from the prior determination of the Plancherel measure by
Template:Harvtxt, giving^{[31]}^{[32]}

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
*m*_{α} = dim 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* Σ_{0} consists of roots α such that α/2 is not a root.

Defining the spherical functions φ_{ λ} as above for λ in , the spherical transform of *f* in C_{c}^{∞}(*K* \ *G* / *K*) is defined by

The **spherical inversion formula** states that

where **Harish-Chandra's c-function** **c**(λ) is defined by^{[33]}

with and the constant *c*_{0} chosen so that **c**(–*i*ρ) = 1 where

The **Plancherel theorem for spherical functions** states that the map

is unitary and transforms convolution by into multiplication by .

## 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 , 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 , 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^{[34]}

and

is the Laplacian on , corresponding to any choice of orthonormal basis (*X*_{i}).

Thus

where

so that *L* can be regarded as a perturbation of the constant-coefficient operator *L*_{0}.

Now the spherical function φ_{λ} is an eigenfunction of the Laplacian:

and therefore of *L*, when viewed as a *W*-invariant function on .

Since *e*^{iλ–ρ} and its transforms under *W* are eigenfunctions of *L*_{0} with the same eigenvalue,
it is natural look for a formula for φ_{λ} in terms of a perturbation series

with Λ the cone of all non-negative integer combinations of positive roots, and the transforms of *f*_{λ} under *W*. The expansion

leads to a recursive formula for the coefficients *a*_{μ}(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 *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 .

It follows that φ_{λ} can be expressed in terms as a linear combination of *f*_{λ} and its transforms under *W*:^{[35]}

Here **c**(λ) is **Harish-Chandra's c-function**. It describes the asymptotic behaviour of φ_{λ} in ,
since^{[36]}

Harish-Chandra obtained a second integral formula for φ_{λ} and hence **c**(λ) using the Bruhat decomposition of *G*:^{[37]}

where *B* = *MAN* and the union is disjoint. Taking the Coxeter element *s*_{0} 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 φ_{λ} initially defined over *K*/*M*

can be transferred to σ(*N*):^{[38]}

Since

for *X* in , the asymptotic behaviour of φ_{λ} can be read off from this integral, leading to the formula:^{[39]}

## Harish-Chandra's c-function

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The many roles of Harish-Chandra's **c**-function in non-commutative harmonic analysis are surveyed in Template:Harvtxt. 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 Template:Harvtxt. 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 ξ_{0}, the constant function 1, in L^{2}(*K*/*M*).^{[40]} Equivalently, since ξ_{0} 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 L^{2}(*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^{[41]}

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^{[42]}

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
*s*_{0}, 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 ξ_{0} is the constant function 1 in L^{2}(*K*/*M*). The cocycle property of the intertwining operators implies a similar multiplicative property for the **c**-functions:

provided

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 Template:Harvtxt. In fact the integral involves only the closed connected subgroup *G*^{α} corresponding to the Lie subalgebra generated by where α lies in Σ_{0}^{+}.^{[43]} 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;^{[6]}^{[44]}

- by directly computing the integral, which can be expressed as an integral in two variables and hence a product of two beta functions.
^{[45]}^{[46]}

This yields the following formula:

where

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

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 (Template:Harvtxt pg. 37).

The theorem was later proved by Template:Harvtxt independently of the spherical inversion theorem, using a modification of his method of reduction to the complex case.^{[47]}

## Rosenberg's proof of inversion formula

Template:Harvtxt 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 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.^{[48]}

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 .^{[49]}^{[50]}

## Schwartz functions

The Harish-Chandra Schwartz space can be defined as^{[51]}

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.^{[6]}^{[7]}^{[52]} Template:Harvtxt 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.

## Notes

- ↑ Template:Harvnb, historical notes on the Plancherel theorem for spherical functions
- ↑ Template:Harvnb
- ↑ Template:Harvnb
- ↑ Template:Harvnb
- ↑ Template:Harvnb
- ↑
^{6.0}^{6.1}^{6.2}Template:Harvnb - ↑
^{7.0}^{7.1}Template:Harvnb - ↑ Template:Harvnb
- ↑ Template:Harvnb, section 21
- ↑ The spectrum coincides with that of the commutative Banach *-algebra of integrable
*K*-biinvariant functions on*G*under convolution, a dense *-subalgebra of . - ↑ The measure class of μ in the sense of the Radon–Nikodym theorem is unique.
- ↑ Template:Harvnb
- ↑ Template:Harvnb
- ↑ Template:Harvnb
- ↑
^{15.0}^{15.1}^{15.2}Template:Harvnb - ↑ Template:Harvnb
- ↑ Template:Harvnb
- ↑ Template:Harvnb
- ↑ Template:Harvnb
- ↑ Template:Harvnb
- ↑ Template:Harvnb
- ↑ These are indexed by highest weights shifted by half the sum of the positive roots.
- ↑ Template:Harvnb
- ↑ Template:Harvnb
- ↑ Template:Harvnb
- ↑ The spherical inversion formula for
*U*is equivalent to the statement that the functions form an orthonormal basis for the class functions. - ↑ Template:Harvnb
- ↑ Template:Harvnb
- ↑ Template:Harvnb
- ↑
*b*(λ) can be written as integral over*A*_{0}where*K*=*K*_{0}*A*_{0}*K*_{0}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*/*K*_{0}. An equivalent computation that arises in the theory of the Radon transform has been discussed by Template:Harvtxt, Template:Harvtxt and Template:Harvtxt. - ↑ Template:Harvnb
- ↑ Template:Harvnb
- ↑ Template:Harvnb
- ↑ Template:Harvnb
- ↑ Template:Harvnb
- ↑ Template:Harvnb
- ↑ Template:Harvnb
- ↑ Template:Harvnb
- ↑ Template:Harvnb
- ↑ Template:Harvnb, Chapter VII
- ↑ Template:Harvnb
- ↑ Template:Harvnb
- ↑ Template:Harvnb
- ↑ Template:Harvnb
- ↑ Template:Harvnb
- ↑ Template:Harvnb
- ↑ The second statement on supports follows from Flensted-Jensen's proof by using the explicit methods associated with Kostant polynomials instead of the results of Mustapha Rais.
- ↑ Template:Harvnb
- ↑ Template:Harvnb
- ↑ Template:Harvnb
- ↑ Template:Harvnb
- ↑ Template:Harvnb

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