4 1. INTRODUCTION
Associated to the quadratic form Q, we define the isotropic cone
C := {x
Rn
\ {0} : Q(x) = 0},
and endow C with the volume form characterized by
dQ = dx1 · · · dxn.
Our object of study is the Hilbert space L2(C) L2(C, dμ) consisting of square
integrable functions on C.
A differential operator P on Rn is said to be tangential to the submanifold C
if P satisfies
(1.1.2) ψ1|C = ψ2|C (Pψ1)|C = (Pψ2)|C
for any smooth functions ψ1,ψ2 defined in neighborhoods of C in
Rn.
Then, we
can ‘restrict’ P to C, and get a differential operator P |C on C.
For instance, the following vector fields are tangential to C:
E :=
n
i=1
xi

∂xi
(the Euler operator),
Xij :=
i j
xi

∂xj
xj

∂xi
(1 i j n),
where we set
j
= 1 or −1 according as 1 j n1 or n1 + 1 j n. This is
because the vector fields E and Xij (1 i j n) are obtained as the differential
of the conformal linear transformation group
CO(Q) := {g GL(n, R) : Q(gx) = cQ(x)
(∀x

Rn)
for some c 0},
which preserves the isotropic cone C.
Let R[x,

∂x
] be the R-algebra of differential operators with polynomial coef-
ficients (the Weyl algebra), namely, the non-commutative ring generated by the
multiplication by
x1,...,xn and the vector fields
∂x1
, . . . ,
∂xn
.
We denote by R[x,

∂x
]C
the subalgebra consisting of operators that are tan-
gential to C. The multiplication by coordinate functions xk clearly satisfies the
condition (1.1.2). Thus, we have seen
xk,E,Xij R x,

∂x
C
(1 k n, 1 i j n).
However, there exist yet other operators which are tangential to C, but are not
generated by xk,E,Xij in the Weyl algebra (see Remark 2.4.9).
Among them are the fundamental differential operators of second order, to be
denoted by P1,...,Pn, which are defined by
(1.1.3) Pj := jxj (2E + n 2)

∂xj
.
Here, is the Laplace–Beltrami operator associated to Q, namely,
:=
n
j=1
j
∂2
∂xj 2

∂2
∂x1 2
+ · · · +
∂2
∂xn1 2

∂2
∂xn1+1 2
· · ·
∂2
∂xn2
.
In the degenerate case n1 = n2 = 1, our operators P1 and P2 take the following
form: we set y1 := x1 + x2, y2 := x1 x2,
P1 + P2 = −4y1
∂2
∂y1
2
, P1 P2 = −4y2
∂2
∂y22
,
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