CHAPTER 1

Introduction

This book is a continuation of a series of our research projects [44, 45, 47, 48,

49, 50]. Our motif is to open up and develop geometric analysis of a single infinite

dimensional representation, namely, the minimal representation π of the indefinite

(even) orthogonal group.

This representation is surprisingly rich in its different models, through which

we have cross-fertilization and interactions with various areas of mathematics such

as conformal geometry and the Yamabe operator, Fourier analysis, ultra-hyperbolic

equations and their conserved quantities, the Kepler problem, holomorphic semi-

groups, and analysis on isotropic cones. Among them, this book is devoted to the

L2-model

(Schr¨ odinger model), for which the local formula was established in a

previous paper [50] with B. Ørsted. The global formula of the whole group action

is the subject of this book.

We have limited ourselves to the very representation π, although some of our

results could be generalized to other settings by the ideas developed here. This

is primarily because we believe that geometric analysis of this specific minimal

representation is of interest in its own right, and might open up an unexpected

direction of research bridging different fields of mathematics, as in the case of the

Weil representation (e.g. [19, 36, 37, 39, 62]).

Bearing this in mind, we will not only

• formalize our main results by means of representation theory,

but also

• formalize our main results without group theory.

We have made effort to expound the theory in a self-contained fashion as much

as possible.

For n = n1 + n2, we denote by Rn1,n2 the Euclidean space Rn endowed with

the flat pseudo-Riemannian structure

ds2

= dx1

2

+ · · · + dxn1

2

− dxn1+1

2

− · · · −

dxn,2

and define the isotropic cone C by

C := {x ∈

Rn

\ {0} : x1

2

+ · · · + xn1

2

− xn1+1

2

− · · · − xn

2

= 0}.

In this book, we will introduce the ‘Fourier transform’ FC on the isotropic cone

C for n even. This transform FC is in some sense the unique and natural unitary

operator on

L2(C),

as it is the case with the Euclidean Fourier transform FRn on

L2(Rn).

Here is a brief guide to the three motivations of this book, with emphasis on

the role of the unitary operator FC.

The first motivation comes from analysis on the isotropic cone C itself. Different

from non-isotropic hypersurfaces (e.g. hyperboloids) in Rn1,n2 , the restriction of ds2

1