8 1. INTRODUCTION

In summary, K4 is a Plancherel type theorem of FC on L2(C), K6 gives its

inversion formula, K7 expresses FC by ‘plane wave’ decomposition, and K8 gives

a Paley–Wiener type theorem for the ‘Schwartz space’ L2(C)∞.

The above formulation brings us naturally to the following program:

Program 1.2.5. Develop a theory of ‘Fourier analysis’ on the isotropic cone

C by means of FC.

We expect that this program could be enhanced by a solid foundation and

concrete formulas of the transform FC.

For this, the first step is to find explicit formulas for the (normalized) joint

eigenfunctions K(x, ξ). We prove that they are given by means of Bessel distri-

butions (see Theorem 1.3.1). In particular, K2 follows readily from the formulas.

The properties K1, K4, K5, and K6 will be proved in Theorem 2.5.2 based on

a representation theoretic interpretation that FC is the ‘unitary inversion oper-

ator’ on

L2(C)

for the minimal representation of the indefinite orthogonal group

O(n1+1,n2+1). By K6, we get the inversion formula just as

FC1 −

= FC, which gives

an explicit solution to the problem of joint eigenfunction expansions (see Problem

1.1.1):

f(x) =

C

(FC f)(ξ)K(x, ξ)dμ(ξ).

The kernel K(x, ξ) is not locally integrable but is a distribution in general. To

see the convergence of the right-hand side (1.2.6), we note that K(x, ξ) depends only

on x, ξ =

∑n

i=1

xiξi (see Theorem 1.3.1). This fact leads us to the factorization

K7 through the (singular) Radon transform R on the isotropic cone C, which is

defined by the integration over the intersection of C with the hyperplane

{x ∈

Rn

: x, ξ = t}.

For a quick summary of the transform R (see Chapter 5 for details), we identify

a compactly supported smooth function f on C with a measure fdμ. It is a tem-

pered distribution on

Rn

(n 2). Then, the Radon transform R of fdμ is defined

by

(1.2.7) Rf(ξ, t) :=

C

f(x)δ( x, ξ − t)dμ(x)

for (ξ, t) ∈

(Rn

\ {0}) × (R \ {0}). The point here is that the integration is taken

over the isotropic cone. In other words, Rf(ξ, t) is obtained by the integration

over submanifolds which are generically of codimension two in Rn. Consequently,

Rf(ξ, t) satisfies the ultra-hyperbolic differential equation of the ξ-variable:

(1.2.8)

n1

j=1

∂2

∂ξj 2

−

n

j=n1+1

∂2

∂ξj2

Rf(ξ, t) = 0.

Next, in order to see the regularity of Rf(ξ, t) at t = 0, we fix ξ. Then, the

intersection of the isotropic cone C with the hyperplane {x ∈

Rn

: x, ξ = t} forms

a one parameter family of submanifolds of codimension two for t = 0, which have

singularities at t = 0. Accordingly, the Radon transform Rf(ξ, t) is not of C∞ class

at t = 0 even for f ∈ C0 ∞(C). The regularity of Rf(ξ, t) at t = 0 is the principal

object of the paper [55], where it is proved that Rf(ξ, t) is [

n−5

2

] times continuously

differentiable at t = 0. Here, [x] denotes the greatest integer that does not exceed