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.)