Analysis on Non-Riemannian Symmetric SpacesHarmonic analysis on Riemannian semisimple symmetric spaces and on special types of non-Riemannian semisimple symmetric spaces are well-established theories. This book presents a systematic treatment of the basic problems on semisimple symmetric spaces and a discussion of some of the more important recent developments in the field. The author's primary contribution has been his idea of how to construct the discrete series for such a space. In this book a fundamental role is played by the ideas behind that construction, namely the duality principle, the orbit picture related to it, and the definition of representations by means of distributions on the orbits. Intended as a text at the upper graduate level, the book assumes a basic knowledge of Fourier analysis, differential geometry, and functional analysis. In particular, the reader should have a good knowledge of the general theory of real and complex Lie algebras and Lie groups and of the root and weight theories for semisimple Lie algebras and Lie groups. |
Contents
Structure and Classification of Symmetric Spaces | 1 |
Harmonic Analysis on Semisimple Symmetric Spaces | 13 |
The Noncompact Riemannian Form Xr of a Semisimple Symmetric Space X | 25 |
The Poisson Transform on a Symmetric Space of the Noncompact Type | 33 |
The Hdorbits on the Boundary of Xr and the Corresponding Representations of G | 43 |
Representations Related to the Closed Hdorbits | 49 |
The Discrete Series for a Semisimple Symmetric Space | 61 |
A Few Final Remarks | 69 |
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Common terms and phrases
0-invariant Cartan a₁ affine symmetric spaces analytic subgroup assume canonical Cartan subspace Chapter closed Hd-orbit Corollary corresponding decomposition defined denote discrete series representation distribution dominant weight eigenfunction eigenspace representations equivalent Example finite finite-dimensional representation Flensted-Jensen follows G₁ G₁/d(G₁ G₁/K₁ Gd/Kd Gd/Pd Harish-Chandra harmonic analysis Hd-finite invariant isomorphic Iwasawa Iwasawa decomposition K-finite K-type K)-module K₁ Lemma Let G Lie group linear Math Matsuki minimal parabolic subgroups modules noncompact Riemannian form noncompact type nontrivial notation orbits Oshima P(ad parabolic subgroup Plancherel formula Plancherel measure Poisson transform positive system proof of Proposition pseudo-Riemannian REMARK representation of G Riemannian symmetric space S(ac satisfies Schlichtkrull semisimple symmetric space simple simply connected spherical function subalgebra subsymmetric space symmetric Lie algebra symmetric triple T₁ Theorem U(gc U₁ unitary representation V₁ vector Vogan W₁ Weyl group
Popular passages
Page 73 - J. ARTHUR, A Paley- Wiener theorem for real reductive groups, Acta Math. 150 (1983), 1-89.
Page 73 - A. BEILINSON AND J. BERNSTEIN, Localisation de g-modules, CR Acad. Sci. Paris 292 (1981), 15-18. 2. A. BEILINSON, J. BERNSTEIN, AND P. DELIGNE, Faisceaux perverse, Ast. 100, 1982. 3. F. BIEN, "Spherical D-Modules and Representations of Reductive Lie Groups," Thesis, MIT, Cambridge, MA, 1986.
Page 73 - P. CARTIER Vecteurs differentiables dans les representations unitaires des groupes de Lie, Sem.