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**Analysis on non-Riemannian symmetric spaces.**
*(English)*
Zbl 0589.43008

Regional Conference Series in Mathematics 61. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-0711-0). x, 77 p. (1986).

The author provides a detailed report on analysis on non-Riemannian symmetric spaces. He begins by reviewing basic structure theory and basic facts about invariant differential operators and their eigenfunctions. There follows a discussion of the ”duality principle” relating a non- Riemannian symmetric space to its Riemannian dual, which is applied to give a new proof of the Paley-Wiener theorem in the Riemannian case. He introduces the Poisson transform and proves that any H-finite joint eigenfunction on a Riemannian symmetric space is the Poisson transform of a distribution on the boundary.

After reviewing the classification of the dual-H-orbits on the Riemannian dual, the author gives a detailed account of the construction of the discrete series, based on his own work and that of Oshima and Matsuki. He concludes with a discussion of representations attached to nonclosed orbits and of the relation between the discrete series to Zuckerman’s derived functor modules.

After reviewing the classification of the dual-H-orbits on the Riemannian dual, the author gives a detailed account of the construction of the discrete series, based on his own work and that of Oshima and Matsuki. He concludes with a discussion of representations attached to nonclosed orbits and of the relation between the discrete series to Zuckerman’s derived functor modules.

Reviewer: W.Rossmann

### MSC:

43A85 | Harmonic analysis on homogeneous spaces |

22E46 | Semisimple Lie groups and their representations |

53C35 | Differential geometry of symmetric spaces |

43A32 | Other transforms and operators of Fourier type |

58J60 | Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) |