Radon and Fourier transform on symmetric spaces Rainer Felix Mathematisch–Geographische Fakult¨ at, Katholische Universit¨at Eichst¨att Ostenstraße 26, D–85072 Eichst¨att, Germany

IN MEMORIAM BOGDAN ZIEMIAN

Introduction. The aim of this lecture is to present a rather streamlined approach to the Radon (horocycle) transform on a Riemannian symmetric space X = G/K of noncompact type. The central point is the proof of the inversion formula. Afterwards I shall point out the strength of this formula and derive from it in a direct way the inversion and the Plancherel formula for the Fourier transform on symmetric spaces. The general theory of Radon and Fourier transform on symmetric spaces is due to Helgason and it can be found in Helgason’s books ([He3 – 6], compare also [He1, 2]). Thus these books are the basis for our expositions and all necessary details can be found there. However, I hope to present some new methods which are slightly different from Helgason’s procedure and which are also aimed at a generalization of the results to a bigger class of spaces: At first, the inversion formula is proved in the origin. (Here, the only tools are Harish–Chandra’s inversion formula for the spherical transform [H– Ch] and the inversion formula for the Mellin transform.) After this is done, the Radon inversion formula in an arbitrary point follows easily by compatibility properties of the occuring operators with the action of the group G. In our studies, the well–known relation between Radon, Fourier and Mellin transform is of crucial importance. 1. Root space decomposition. Let G be a real connected semisimple Lie group with finite centre and let g be the Lie algebra of G. Let θ be a Cartan involution and let g = k ⊕ p be the corresponding Cartan decomposition of g into the ± 1–eigenspaces of θ. The analytic subgroup K of G corresponding with k is compact. Let a be a maximal commutative subspace of p. For any λ in the (real) dual space a∗ of a we put gλ := {X ∈ g | [H, X] = λ(H)X for all H ∈ a}. λ ist called a root of (g, a) if λ 6= 0 and gλ 6= {0}. We have the root space decomposition g = g0 ⊕

M

gλ ,

λ∈Σ

1

g0 = a ⊕ m,

where Σ is the set of all roots and m is the centralizer of a in k. The root spaces satisfy the condition [gλ , gµ ] ⊆ gλ+µ for all λ, µ ∈ Σ. 2. Positive roots. The set of regular elements in a is defined by a0 := {H ∈ a | λ(H) 6= 0 for all λ ∈ Σ}. Clearly, a0 is the complement of finitely many hyperplanes in a. The connected components of a0 are the Weyl chambers. Let M and M 0 the centralizer and the normalizer of a in K, respectively. Obviously, M is normal in M 0 . The quotient group W := M 0 /M is the Weyl group; it acts transitively on the Weyl chambers. Now we fix a Weyl chamber a+ . We call a root λ positive if λ(H) > 0 for all H ∈ a+ . Let Σ+ be the set of positive roots and let Σ− := {−λ | λ ∈ Σ+ } = Σ − Σ+ . We put n :=

M

¯ := and n

gλ

λ∈Σ+

M

gλ .

λ∈Σ−

¯ are nilpotent subalgebras of g by virtue of the relation [gλ , gµ ] ⊆ gλ+µ . n and n ¯ are normalized Clearly, each root space gλ is Ad(M )–invariant, and therefore n and n ∗ by M. We define an element ρ of a by ρ(H) :=

1 Tr ad(H)|n . 2

Let A be the analytic subgroup of G corresponding with a. For elements s = σ + iτ in the complexification a∗c of a∗ and for a ∈ A we put as := es(Log a) . ¯ the analytic subgroups of G corresponding with n and Furthermore, let N and N ¯ ¯, respectively. We have N = θN where θ is the automorphism of G lifted from the n Cartan involution θ of g. Obviously, N is normalized by A such that the semidirect product S := AN = N A is a solvable subgroup of G. An easy calculation shows that a−2ρ is the modulus of the automorphism n 7→ a n a−1 of N ; i.e. Z

Z −1

−2ρ

f (ana )dn = a N

f (n)dn, N

2

f ∈ Cc (N ),

where dn is the Haar measure on N. 3. Iwasawa decomposition. Our Lie algebra g decomposes into the direct sum g = n ⊕ a ⊕ k. From this we can derive the Iwasawa decomposition of G : G = N A K. The mapping N × A × K → G,

(n, a, k) 7→ nak

is a diffeomorphism. We introduce the projection π : G → A, defined by π(g) := a where g = nak, and we put A(g) := Log π(g). By passage to the inverse element we see that any g ∈ G can also be written as g = k 0 a0 n0 with k 0 ∈ K, a0 ∈ A, n0 ∈ N. We put H(g) := Log a0 . Clearly, we have H(g) = −A(g −1 ). Obviously, π(g) and A(g) are K–right–invariant and H(g) is K–left–invariant. The Haar measure dg on G can be calculated from the Haar measures dn, da and dk on N, A and K, respectively, by the formula Z

Z Z Z f (g)dg =

G

f (ank)dk dn da, A

N

f ∈ Cc (G).

K

Keeping in mind that a−2ρ is the modulus of the automorphism n 7→ ana−1 of N, we get also the following representation of dg : Z

Z Z Z f (nak)a−2ρ dk da dn.

f (g)dg = G

Z

N

Furthermore, using the relation Z

A

K

Z

G

Z Z Z

G

f (kan)a2ρ dn da dk.

f (g)dg = G

f (g −1 )dg, we receive

f (g)dg =

K

A

N

4. Normalization of Haar measures. Of course, we shall normalize the measures dk and dm on the compact groups K and M, respectively, such that the total measure is 1. In order to normalize the measure da we notice that the Killing form is positive 3

definit on p, and thus it gives Euclidean measures on a and a∗ . By multiplying these measures with the constant (2π)−l/2 , where l = dim a, we get measures dH and dλ on a and a∗ , respectively, which are called regularly normalized. Now we define the Haar measure da by the equation Z

Z h(a)da :=

h(Exp H)dH,

h ∈ Cc (A).

a

A

The Mellin transform M[h] of a function h on A is defined on a∗c by Z h(a)as da ,

M[h](s) =

s ∈ a∗c .

A

Now let us normalize the measure dn : First we normalize the Haar measure d¯ n on ¯ N such that Z e−2ρ(H(¯n)) d¯ n=1. ¯ N

This normalization of d¯ n gives us a normalization of dn by putting dn := θ(d¯ n) . After this normalization conventions we receive a normalization of the Haar measure dg on G by the formulae in Section 3. 5. Symmetric space. The basic object of our study is the symmetric space X := G/K. In view of the Iwasawa decomposition, X can be identified with S = N A = AN ∼ = A × N by the mapping S → X , na 7→ na · o where o is the origin in X . The measure dg˙ on G/K defines a G–invariant measure dx on X which coincides with the left Haar measure da dn on S by the identification X∼ =S. Because the mappings π(g) and A(g) on G defined in Section 3 are K–right– invariant, they can be viewed as mappings π(x) and A(x) on X = G/K . Taking into account that N is normalized by M we have the equations π(m · x) = π(x) and A(m · x) = A(x) ,

x ∈ X, m ∈ M .

6. Horocycles. The horocycles in X are the orbits of all subgroups conjugate to N ; i.e. the space Ξ of horocycles consists of all subsets ξ of X of the form

4

ξ = gN g −1 · x ,

g ∈ G, x ∈ X .

G acts naturally on Ξ . The horocycle E := N · o is to be considered as the standard horocycle. Each horocycle can be written as ka · E, where k ∈ K, a ∈ A . Thus G acts transitively on Ξ . The stabilizer of E for this action is M N = N M . Therefore, Ξ can be identified with G/M N . This identification gives rise to a structure of manifold on Ξ . The mapping ˙ a) 7→ ka · E (k,

K/M × A → Ξ , is a diffeomorphism.

The measure dg˙ on G/M N gives a G–invariant measure dξ on Ξ which corresponds to the measure a2ρ da dk˙ ∧

by the identification Ξ = K/M × A . 7. Radon transform. By the bijection n 7→ n · o from N onto the standard horocycle E = N · o the Haar measure dn induces a measure µ on E which is M N –invariant. Thus we can define the Radon transform Rf of a function f on X by integration ∧ over the horocycles. By the identification Ξ = K/M × A we define Rf as a function on K/M × A : Z ˙ a) := Rf (k,

Z f (ka · z)dµ(z) =

E

f (kan · o)dn ,

˙ a) ∈ K/M × A . (k,

N

It is easily seen that R commutes with the action of G. That means the following: If f (x) and φ(ξ) are functions on X and Ξ, respectively, then, for g ∈ G, we define the functions g f (x) := f (g · x) and g φ(ξ) := φ(g · ξ), and the equation R(g f ) = g (Rf ) ˙ a) holds. Particularly, if f is K–invariant, then Rf is also K–invariant; i.e. Rf (k, ˙ Furthermore, if f belongs to Cc (X) or D(X), then Rf (k, ˙ a) does not depend on k. ˙ exists for all (k, a) ∈ Ξ, and φ := Rf belongs to Cc (Ξ) or D(Ξ), respectively. 8. Dual transform. Now we want to define the dual transform R∗ of R, which associates to a function φ(ξ) a function R∗ φ(x) such that for functions f (x) the equation

5

Z

Z f (x)R∗ φ(x)dx

Rf (ξ)φ(ξ)dξ = Ξ

X

holds. To carry out this we compute the left hand side of this equation: Z

Z

Z Rf (ka · E)φ(ka · E)a2ρ da dk˙

Rf (ξ)φ(ξ)dξ = Ξ

A Z Z

K/M Z

˙ a)a2ρ dn da dk˙ f (kan · o)φ(k,

=

A N Z Z Z

K/M Z

˙ π(ank 0 · o))π(ank 0 · o)2ρ dk 0 dn da dk˙ f (kank 0 · o)φ(k,

= A Z

K/M Z

N

K

˙ π(g · o))π(g · o)2ρ dg dk˙ f (kg · o)φ(k,

= G Z

K/M Z

˙ π(k −1 g · o))π(k −1 g · o)2ρ dg dk˙ f (g · o)φ(k,

= G

K/M Z

=

Z ˙ π(k −1 · x))π(k −1 · x)2ρ dk˙ dx. φ(k,

f (x) X

K/M

This equation suggests to define R∗ φ by Z ˙ π(k −1 · x))π(k −1 · x)2ρ dk, ˙ φ(k,

∗

R φ(x) :=

x ∈ X.

K/M

This integral admits a geometric interpretation: For a given point x ∈ X the compact set ˙ π(k −1 · x)) | k˙ ∈ K/M } {(k, corresponds just to the set Ξx of all ξ ∈ Ξ = ˆ K/M × A, which contain the point x, and the measure Z ˙ π(k −1 · x))π(k −1 · x)2ρ dk˙ φ(k,

φ 7→ K/M

on Ξx is just the normalized Gx –invariant measure, where Gx is the stabilizer of x acting on Ξx . The dual transform commutes with the action of G, too; i.e. 6

R∗ (g φ) = g (R∗ φ),

g ∈ G.

This can be easily derived from the G–invariance of dξ together with the relation between R and R∗ . 9. Spherical transform. Now we shall focus our attention on the functions π(x)s , s = σ + iτ ∈ a∗c . These functions are joint eigenfunctions of the set D(X) of G–invariant differential operators on X . We receive the spherical functions by symmetrizing the functions π(x)s with respect to the action of K . In view of the spherical transform and its inversion we confine ourself to the case that s = iλ + ρ, where λ ∈ a∗ and ρ is our modulus function defined in Section 2. Thus we put for any λ ∈ a∗ Z π(k −1 · x)iλ+ρ dk ,

ϕλ (x) :=

x∈X.

K

Of course, the spherical functions ϕλ can also be viewed as K–biinvariant functions on G . The spherical transform of a K–invariant function F on X (which can also be viewed as a K–biinvariant function on G) is defined by Z F˜ (λ) :=

Z F (x)ϕλ (x)dx =

F (x)ϕ−λ (x)dx ,

X

λ ∈ a∗ .

X

It is inverted by means of Harish-Chandra’s c–function which is given by Z e−(iλ+ρ)(H(¯n)) d¯ n.

c(λ) = ¯ N

The function c(λ)−1 has slow growth on a∗ . The inversion formula is Z F (x) = w

F˜ (λ)ϕλ (x)|c(λ)|−2 dλ ,

−1

x∈X,

a∗

where w is the order of the Weyl group. 10. Fourier transform. Now let f (x) be a function on X which is not necessarily K–invariant. We symmetrize this function and get a K–invariant function Z f (k −1 · x)dk .

F (x) := K

7

Next we compute the spherical transform of F : Z Z F˜ (λ) =

Z f (k

X Z K

=

−1

· x)ϕ−λ (x)dkdx

=

Z

f (x) X

f (x)ϕ−λ (x)dx XZ

π(k

−1

−iλ+ρ

· x)

Z f (x)π(k −1 · x)−iλ+ρ dxdk˙ .

dkdx =

K

K/M X

The inner integral Z ˙ := Ff (λ, k)

f (x)π(k −1 · x)−iλ+ρ dx,

˙ ∈ a∗ × K/M, (λ, k)

X

is called the Fourier transform of f. The Fourier transform is closed related with the Radon transform. In fact, we have Z ˙ = Ff (λ, k)

Z Z f (k · x)π(x)

−iλ+ρ

f (kan · o)π(an · o)−iλ+ρ dn da

dx =

X Z

A

N

˙ a)a−iλ+ρ da = M[Rf ](k, ˙ −iλ + ρ), Rf (k,

= A

˙ s) is the “partial” Mellin transform of Rf, i.e. the Mellin transform where M[Rf ](k, ˙ a). Thus we have the crucial equation of the function A 3 a 7→ Rf (k, ˙ = M[Rf ](k, ˙ −iλ + ρ). Ff (λ, k) This equation shows that Ff is a Schwartz function on a∗ ×K/M whenever f ∈ D(X). 11. Inversion operator. For inverting the Radon transform we need an operator Λ on A which transforms a function h ∈ D(A) to a function Λh on A. This operator is derived from the c–function using the Mellin transform. Observing the properties of the Mellin transform and taking into account that c(λ)−1 has slow growth, we can define such an operator Λ by the equation c(λ)−1 · M[h](−iλ + ρ) = M[Λh](−iλ + ρ). Λ is also an operator on Ξ = ˆ K/M × A by acting only on the A–component. If Λ is viewed as an operator on Ξ it is G–invariant; i.e. Λ(g φ) = g (Λφ) , g ∈ G, φ ∈ D(Ξ). In fact, putting s := −iλ + ρ we have for any k ∈ K the relation 8

˙ s) = a−s M[φ](k˙ 1 , s) M[g φ](k, 1 where gk = k1 a1 n1 with k1 ∈ K, a1 ∈ A, n1 ∈ N. We conclude

˙ s) = c(λ)−1 · M[g φ](k, ˙ s) = c(λ)−1 · a−s M[φ](k˙ 1 , s) = a−s M[Λφ](k˙ 1 , s) M[Λ(g φ)](k, 1 1 ˙ s). = M[g (Λφ)](k, ¯ In the same way, starting by c(−λ)−1 , we define an operator Λ. ˜ := w−1 ΛΛ ¯ will In the next section we shall see that the G–invariant operator L ˜ the inversion give us the inversion formula for the Radon transform. Let us call L operator. 12. Radon inversion formula. The Radon inversion formula enables us to recover a function f ∈ D(X) from its Radon transform Rf. We shall achieve this formula by using the inversion formulae for the sperical and the Mellin transform. First we prove the Radon inversion formula in the origin. Coming back to the situation in Section 10, we get a K–invariant function F by symmetrizing f and conclude: Z f (o) = F (o) = w Z

Z

= w−1

Z

Z −2 ˙ ˙ Ff (λ, k)|c(λ)| dk dλ

F˜ (λ)|c(λ)|−2 dλ = w−1

−1 a∗

a∗ K/M

˙ −iλ + ρ)c(−λ)−1 c(λ)−1 dλ dk˙ M[Rf ](k,

∗ K/M Z aZ

= w

−1

Z ˙ −iλ + ρ)dλ dk˙ = ¯ M[ΛΛRf ](k,

K/M a∗

˙ e)dk˙ ˜ (k, LRf K/M

∗˜

= R LRf (o). Now we receive immediately the general inversion formula ˜ (x), f (x) = R∗ LRf

x ∈ X,

˜ and R∗ are G–invariant. Namely, if x = g · o we have by the fact that R, L ˜ (x). ˜ )(o) = R∗ LRf ˜ g f )(o) = g (R∗ LRf f (x) = g f (o) = R∗ LR(

9

13. Inversion and Plancherel formula for the Fourier transform. In this last section we shall employ the Radon inversion formula to derive the inversion and Plancherel formula for the Fourier transform. First, given f, h ∈ D(X) we develop the following equation by using the Parseval– Plancherel equality for the Mellin transform: Z

Z

Z ∗˜

h(x)f (x)dx = XZ

X

˜ (ξ)dξ Rh(ξ)LRf

h(x)R LRf (x)dx = Z

Ξ

˙ a)LRf ˙ a)a2ρ da dk˙ ˜ (k, Rh(k,

= A Z

K/M Z

˙ −iλ + ρ)M[LRf ˙ −iλ + ρ) dλ dk˙ ˜ ](k, M[Rh](k,

= ∗ K/M aZ

= w

Z ˙ −iλ + ρ)|c(λ)|−2 M[Rf ](k, ˙ −iλ + ρ) dλ dk˙ M[Rh](k,

−1 ∗ K/M Z aZ

−2 ˙ ˙ ˙ Fh(λ, k)Ff (λ, k)|c(λ)| dλ dk.

= w−1 K/M a∗

By putting h = f the Plancherel formula for the Fourier transform (except of the surjectivity statement) results: Z

Z Z 2

|f (x)| dx = w

˙ 2 |c(λ)|−2 dλ dk. ˙ |Ff (λ, k)|

−1 K/M a∗

X

To receive the inversion formula in a given point y ∈ X we put h = hν , where (hν )ν∈N is a sequence Rof functions converging to the Dirac function δy ; i.e. we take functions hν ≥ 0 with hν (x)dx = 1 such that the support of hν shrinks to arbitrarily small X

neighbourhoods of y for ν → ∞. Now the limit process ν → ∞ gives the Fourier inversion formula: Z f (y) = w

Z

−1

˙ |c(λ)|−2 dλ dk˙ . π(k −1 · y)iλ+ρ Ff (λ, k)

K/M a∗

References [H–Ch] Harish–Chandra: Spherical functions on a semisimple Lie group, I, II. Amer. J. Math. 80, 241–310, 553–613 (1958) 10

[He1] Helgason, S.: Radon–Fourier transforms on symmetric spaces and related group representations. Bull. Amer. Math. Soc. 71, 757–763 (1965) [He2] Helgason, S.: A duality for symmetric spaces with applications to group representations. Adv. Math. 5, 1–154 (1970) [He3] Helgason, S.: Differential geometry, Lie groups, and symmetric spaces. New York, San Francisco, London: Academic Press 1978 [He4] Helgason, S.: The Radon Transform. Progress Math. 5. Boston–Basel–Stuttgart: Birkh¨auser 1980 [He5] Helgason, S.: Groups and geometric analysis. Integral geometry, invariant differential operators, and spherical functions. Orlando etc.: Academic Press 1984 [He6] Helgason, S.: Geometric analysis on symmetric spaces. Providence, RI: American Mathematical Society 1994

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