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Orbital symmetric spaces and finite multiplicity. (English) Zbl 0843.22018
The author studies algebraic symmetric spaces. Let \(G\) be a real algebraic group, \(\sigma\) an involution of \(G\) and let \(H\) be the set of fixed points of \(\sigma\). In terms of the orbit method the author gives a necessary and sufficient condition for the quasi-regular representation \(\tau = \text{ind}^G_N 1\) to be irreducible and to be a finite sum of irreducibles. Then he proves that any algebraic symmetric space has finite multiplicity (i.e. \(\tau\) has finite multiplicity) and examines the boundedness of the multiplicity. To set up the inductive scheme, he slightly generalizes the situation. Let \(Z\) be a unipotent connected \(\sigma\)-invariant central subgroup of \(G\), and let \(\chi\) be a unitary character of \(HZ\) satisfying \(\sigma \chi = \overline{\chi}\). Then \(\tau\) is replaced by the induced representation \(\tau = \text{ind}^G_{HZ} \chi\). – The method of proof is an amalgam of the Mackey machine and the orbit method based on the works of L. Auslander and B. Kostant [Invent. Math. 14, 255-354 (1971; Zbl 0233.22005)], L. Corwin, F. P. Greenleaf and G. Grélaud [Trans. Am. Math. Soc. 304, 549-583 (1987; Zbl 0629.22005)], M. Duflo [Ann. Sci. Éc. Norm. Supér., IV. Sér. 5, 71-120 (1972; Zbl 0241.22030)] and on a series of the author’s previous researches [Pac. J. Math. 140, 117-147 (1989; Zbl 0645.43010); ibid. 159, No. 2, 351-377 (1993; Zbl 0798.22005); “Representation theory of Lie groups and Lie algebras”, 120-139, World Scientific, Singapore (1992)].

22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
43A85 Harmonic analysis on homogeneous spaces
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