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.
Reviewer: W.Rossmann


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.)