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
Previous Page Next Page