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  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.
ix
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