Introduction

Harmonie analysis on Symmetrie spaees is for me a very inspiring combination

of analysis, geometry and algebra. In this book I shall try to present this subject

with special emphasis on those pseudo-Riemannian Symmetrie spaees whieh have

a semisimple group of isometries. We shall call these the semisimple Symmetrie

spaees.

Harmonie analysis on Riemannian semisimple Symmetrie spaees is very well

established, primarily through the work of H. Weyl, E. Cartan, Harish-Chandra

and S. Helgason.

Among the non-Riemannian semisimple Symmetrie Spaces are, for example, the

noncompact semisimple groups and the hyperbolic spaees. For these special

examples of non-Riemannian Symmetrie spaees there is also a well-established

harmonic analysis. However, for the general semisimple Symmetrie Spaces,

harmonic analysis is far less developed, and many basic questions have not yet

found a final answer.

My own contribution to this subject is primarily the idea of how to construct

the discrete series for such a space. I hope I am excused for puttihg some

emphasis on this aspect. In [c], where I first presented the construction, I tried to

show that the construction is very elementary and direct. In this book I have

chosen to let the general ideas behind the construction play a fundamental

role—that is, the duality principle and the orbit picture related to it and also the

definition of representations by means of distributions on the orbits. At the same

time I have tried to give a rather systematic treatment of the basic problems in

harmonic analysis on Symmetrie spaees and to discuss some of the more im-

portant recent developments in the theory.

There are a few new results in the text. In Example Â of Chapter III there is a

new and simple proof of the Paley-Wiener theorem for Riemannian Symmetrie

spaees of the noncompact type. In §3 of Chapter IV it is proved that any

"//-finite" Joint eigenfunction on a Riemannian Symmetrie space is the Poisson

transform of a distribution on the boundary. This result implies that we, to a large

extent, can avoid mentioning hyperfunctions in our construction of representa-

tions.

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