Contents
Acknowledgments vii
Introduction ix
I. Structure and Classification of Symmetrie Spaces 1
1 .Affine Symmetrie Spaces 1
2. Reductive Symmetrie Spaces 3
3.Classification of irreducible semisimple Symmetrie Spaces and examples 5
4. Symmetrie triples—association and duality 9
Notes 11
II. Harmonie Analysis on Semisimple Symmetrie Spaces 13
1 .Invariant differential Operators 14
2.The Plancherel formula and eigenspace representations 16
3.Examples 19
Notes 23
III. The Noncompact Riemannian Form
Xr
of a Semisimple Symmetrie
Space X 25
1 .The duality principle 25
2.Examples 27
Notes 31
IV. The Poisson Transform on a Symmetrie Space of the Noncompact Type 33
l.The Poisson transform 33
2.Helgason's conjeeture 34
3.Applications to the non-Riemannian case 37
Notes 40
V. The //^-Orbits on the Boundary and the Corresponding Representations
of G 43
1 .Classification of the T/Sorbits on Gd/Pd 44
2.Examples 45
3.Eigenspace representations in SX{X) determined by the i/^-orbits 46
Notes 47
í
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