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Le théorème de Paley-Wiener invariant pour les groupes de Lie réductifs. II. (The invariant Paley-Wiener theorem for reductive Lie groups. II). (French) Zbl 0724.22012
Let G be a reductive Lie group. The authors assume also that G is the set of real points of a reductive connected algebraic group defined over \({\mathbb{R}}\). Fix a maximal compact subgroup K of G. Let MAN be a cuspidal parabolic subgroup of G. A representation of G is called basic if it is unitarily induced by a representation \(\delta \otimes e^{\nu}\otimes 1\) of MAN where \(\delta\) is a limit of discrete series representations of M, \(e^{\nu}\) is a character of A. Denote by \(C_ c^{\infty}(G,K)\) the space of compactly supported \(C^{\infty}\)-functions on G which are K- finite with respect to left and right action of G. The main theorem characterizes the traces in the basic representations of the functions in \(C_ c^{\infty}(G,K)\). When \(\delta\) ranges only over the discrete series, such theorem was proved in the first part of this work [see Invent. Math. 77, 427-453 (1984; Zbl 0584.22005)]. Another theorem of the paper reviewed concerns the surjectivity of Harish-Chandra homomorphisms for the limits of discrete series. It extends a theorem proved by the second author [Ann. Sci. Ec. Norm. Supér., IV. Sér. 17, 117-156 (1984; Zbl 0582.22009)] for the case of discrete series.

MSC:
22E46 Semisimple Lie groups and their representations
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
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