1.2. ‘FOURIER TRANSFORM’ FC ON THE ISOTROPIC CONE C 5

see Remark 2.4.10. In general, these operators P1,...,Pn satisfy the following

properties (see Theorem 2.4.1):

P1 PiPj = PjPi for any 1 ≤ i, j ≤ n.

P2 Pj ∈ R[x,

∂

∂x

]C

for any 1 ≤ j ≤ n.

P3 The induced differential operators Pj|C on C0 ∞(C) extend to self-adjoint oper-

ators on the Hilbert space L2(C).

P4 (P1 2 + · · · + Pn1 2 − Pn1+1 2 − · · · − Pn)|C 2 = 0.

P5 The Lie algebra generated by xi, Pi (1 ≤ i ≤ n) contains the vector fields

E, Xij (1 ≤ i j ≤ n).

From now on, we simply write Pj for Pj|C . Thus, we have commuting self-

adjoint, second-order differential operators P1,...,Pn on

L2(C).

I. Todorov drew

our attention that the fundamental differential operators P1,...,Pn appeared also

in [5, (3.4)] under the name “interior derivatives”.

We are brought naturally to the following:

Problem 1.1.1. 1) Find joint eigenfunctions of the differential operators

P1,...,Pn on the isotropic cone C.

2) Given a function f on C, find an explicit expansion formula of f into joint

eigenfunctions of P1,...,Pn.

1.2. ‘Fourier transform’ FC on the isotropic cone C

In this book, we shall give a solution to Problem 1.1.1 by introducing a unitary

operator FC on

L2(C).

To elucidate the operator FC, let us consider first much simpler operators

pj := −

√

−1

∂

∂xj

(1 ≤ j ≤ n)

in place of Pj. Then, p1,...,pn form a commuting family of differential operators

which extend to self-adjoint operators on

L2(Rn).

Analogously to Problem 1.1.1,

consider the question of finding the explicit eigenfunction expansion for the op-

erators p1,...,pn. Then, as is well-known, this is done by using the (Euclidean)

Fourier transform F ≡ FRn on

Rn.

In what follows, we normalize FRn as

(1.2.1) FRn u(ξ) :=

1

(2π)

n

2 Rn

u(x)e

√

−1 x,ξ

dx,

where x, ξ =

∑n

i=1

xiξi and dx = dx1 · · · dxn. We note that the signature of the

power here is opposite from the usual convention. Obviously, the kernel

k(x, ξ) :=

1

(2π)

n

2

e

√

−1 x,ξ

of the Fourier transform FRn is real analytic on the direct product space Rn × Rn.

We recall the following key properties of the Euclidean Fourier transform:

F1 pj k(x, ξ) = ξj k(x, ξ).

F2 k(x, ξ) = k(ξ, x).

F3 FRn (C0

∞(Rn))

⊂

C∞(Rn)

∩

L2(Rn).

F4 FRn extends to a unitary operator on L2(Rn).

F5 FRn ◦ xj = pj ◦ FRn ,

FRn ◦ pj = −ξj ◦ FRn .

F6 (FRn 2 u)(x) = u(−x), FRn 4 = id.