1.1. DIFFERENTIAL OPERATORS ON THE ISOTROPIC CONE 3
simple group Mp(n, R) O(n1 + 1,n2 + 1)
(type C) (type D)
Minimal representation Weil representation π
L2-model L2(Rn) L2(C)
(Schr¨ odinger model)
unitary inversion e

−1nπ
4
FRn FC
The third motivation comes from special functions. We note that the isotropic
cone C is so small that the group G = O(n1+1,n2+1) cannot act on C continuously
and non-trivially. This feature is reflected by the fact that the Gelfand–Kirillov
dimension of the representation of G on
L2(C)
attains its minimum amongst all
infinite dimensional representations of G. Thus, the representation space
L2(C) is
extremely ‘small’ with respect to the group G. In turn, we could expect a very
concrete theory of global analysis on C by using abundant symmetries of the group
G or its Lie algebra.
It turns out that special functions in the Schr¨ odinger model L2(C) arise in a
somewhat different way from the well-known cases such as analysis on symmetric
spaces (e.g. [32]) or its variants. For instance, the Casimir operator of K acts on
L2(C) as a fourth differential operator.
In this book, we encounter many classically known special functions (e.g. Bessel
functions, Appell’s hypergeometric functions, Meijer’s G-functions, etc.). Special
functions are a part of our method for the analysis of the minimal representation,
and conversely, by decomposing the operator FC we provide a representation the-
oretic proof of [inversion, Plancherel, . . . ] formulas of special functions including
Meijer’s G-functions.
Encouraged by a suggestion of R. Stanton, we have decided to write a con-
siderably long introduction. What follows is divided into three parts according
to the aforementioned three motivations and new perspectives. In Sections 1.1–
1.3, we state key properties of the involutive unitary operator FC on
L2(C)
from
analytic perspectives, in comparison with the well-known case of the Euclidean
Fourier transform FRn on
L2(Rn).
Sections 1.4–1.8 give representation theoretic
perspectives, and we explain the role of FC in the Schr¨ odinger model of the mini-
mal representation of the indefinite orthogonal group in comparison with the role
of FRn for the Weil representation. Thus, we compare FC again with FRn , and cor-
respondingly, the simple Lie algebra o(n1 + 1,n2 + 1) with sp(n, R). In Section 1.9,
we give a flavor of the interactions of the analysis on the minimal representations
with special functions.
1.1. Differential operators on the isotropic cone
Consider an indefinite quadratic form on Rn = Rn1+n2 :
(1.1.1) Q(x) := x1
2
+ · · · + xn1
2
xn1+1
2
· · · xn1+n2
2
.
Throughout the Introduction, we assume n1,n2 1 and n = n1 + n2 is an even
integer greater than two. (From Chapter 2, we will use the following notation:
n1 = p 1, n2 = q 1.)
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